ms. wilson's math classes - home - state whether each sentence … · 2018. 9. 9. · floor...
TRANSCRIPT
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
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Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 2
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 3
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 4
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 5
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 6
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 7
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 8
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 9
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 10
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 11
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 12
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 13
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 14
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 15
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 16
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 17
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 18
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 19
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 20
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 21
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 22
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 23
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 24
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 25
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 26
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 27
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 28
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 29
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 30
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 31
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 32
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 33
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 34
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 35
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 36
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 37
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 38
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 39
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
eSolutions Manual - Powered by Cognero Page 40
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If false , replace the underlined term to make a true sentence.
1. The factored form of a quadratic equation is where a 0 and a, b, and c are integers.
SOLUTION: False, standard form
2. The graph of a quadratic function is called a parabola.
SOLUTION: true
3. The vertex form of a quadratic function is .
SOLUTION: false
4. The axis of symmetry will intersect a parabola in one point called the vertex.
SOLUTION: true
5. A method called FOIL method is used to make a quadratic expression a perfect square in order to solve the related equation.
SOLUTION: False, completing the square method
6. The equation is known as the discriminant.
SOLUTION: False, Quadratic Formula
7. The number 6i is called a pure imaginary number.
SOLUTION: true
8. The two numbers 2 + 3i and 2 – 3i are called complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function.
9.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 12.
The x-coordinate of the vertex is .
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
10.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function. The y-intercept is 15.
The x-coordinate of the vertex is given by .
So:
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is .
b.
c. The graph of the quadratic function is:
11.
SOLUTION:
a.
To find the y-intercept, substitute x = 0 in the function. The y intercept is –5.
The x-coordinate of the vertex is given by .
Substitute the values a and b.
The line of symmetry passes through the vertex. Therefore, the equation of the axis of symmetry is .
b.
c.
12.
SOLUTION: a. To find the y-intercept, substitute x = 0 in the function.The y-intercept is –1.
The x-coordinate of the vertex is given by .
Equation of the axis of the symmetry is x = 2. b.
c. The graph of the quadratic function is:
Determine whether each function has a maximum or minimum value and find the maximum or minimum value. Then state the domain and range of the function.
13.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. The y-coordinate of the vertex is 1.25. Therefore, the maximum value is 1.25. Domain = {all real numbers}
14.
SOLUTION: The sign of a is negative so the function has a maximum value.
The maximum value occurs at the vertex. Substitute in .
So, the maximum value is
. Domain = {all real numbers}
15. BUSINESS Sal’s Shirt Store sells 100 T-shirts per week at a rate of $10 per shirt. Sal estimates that he will sell 5 less shirts for each $1 increase in price. What price will maximize Sal’s T-shirt income?
SOLUTION: Let x be the price increase, in dollars, and let P(x) be the total income.
The maximum occurs at the vertex.
Use to find the x-coordinate of the vertex.
The price should be increased $5 to $15. Substitute 5 for x in (100 – 5x).
The store should sell 75 T-shirts at $15 each to maximize the income.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
16.
SOLUTION:
Graph the function y = x2 – x – 20.
The graph intersects the x-axis at –4 and 5. So, the roots of the equation are x = –4 and x = 5.
17.
SOLUTION:
Graph the function .
The graph intersects the x-axis at –1 and .
So, the roots of the equation are x = –1 and .
18.
SOLUTION: Graph the function.
The graph intersects the x-axis between –2 and –1, and between 2 and 3. So the roots of the equation are between –2 and –1 and between 2 and 3.
19. BASEBALL A baseball is hit upward at 120 feet per second. Use the formula , where h(t) is the
height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. Ignoring
the height of the ball when it was hit, how long does it take for the ball to hit the ground?
SOLUTION:
Substitute h = 0 and in the equation, and solve for t.
Since t = 0 represents the time the when it was hit, t = 0 is inadmissible.Therefore, it takes 7.5 seconds for the ball to hit the ground.
Write a quadratic equation in standard form with the given roots.
20. 5, 6
SOLUTION:
21. –3, –7
SOLUTION:
22. –4, 2
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
Solve each equation by factoring.
26.
SOLUTION:
The solution set is {–3, 4}.
27.
SOLUTION:
The solution set is .
28.
SOLUTION:
The solution set is .
29. Find x and the dimensions of the rectangle below.
SOLUTION:
The area of a rectangle is given by where l is the length and w is the width.
x cannot be negative. Therefore, x = 12. The dimensions of the rectangle are 14 feet by 9 feet.
Simplify.
30.
SOLUTION:
31. (2 – i) + (13 + 4i)
SOLUTION: Add the real and the imaginary parts separately.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
33. (6 + 5i)(3 – 2i)
SOLUTION:
34. ELECTRICITY The impedance in one part of a series circuit is 3 + 2j ohms, and the impedance in the other part of the circuit is 4 – 3j ohms. Add these complex numbers to find the total impedance in the circuit.
SOLUTION: Add the complex numbers.
The total impedance in the circuit is 7 – j ohms.
Solve each equation.
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.
40.
SOLUTION: To find the value of c, divide the coefficient of x by 2, and square it.
Substitute the value of c in the trinomial.
41.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
42.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
43.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
44.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square it.
Substitute the value of c in the trinomial.
45.
SOLUTION: To find the value of c, divide the coefficient of x by 2 and square the result.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
46.
SOLUTION:
The solution set is {–1, 7}.
47.
SOLUTION:
The solution set is .
48.
SOLUTION:
The solution set is .
49.
SOLUTION:
50. FLOOR PLAN Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280 square feet. What are the dimensions of his living room?
SOLUTION: Let x feet be the width of the room. So, the length is x + 6 feet. The area of a rectangle is given by , where l is length and w is the width. Therefore:
x cannot be negative. So, x = 14. Therefore, the width of the room is 14 feet, and the length is 20 feet.
Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula.
51.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has one real root.
c.
Substitute the values of a, b, and c.
The solution set is {–5}.
52.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is {–8, 4}.
53.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is greater than zero and not a perfect square, the quadratic equation has 2 irrational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
54.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is .
55.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
56.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic equation has 1 real rational root.
c.
Substitute the values of a, b, and c.
The solution set is .
57.
SOLUTION:
a.
Substitute the values of a, b, and c.
b. Since the discriminant is less than zero, the quadratic equation has 2 complex roots.
c.
Substitute the values of a, b, and c.
The solution set is .
58. PHYSICAL SCIENCE Lauren throws a ball with an initial velocity of 40 feet per second. The equation for the
height of the ball is h = –16t2 + 40t + 5, where h represents the height in feet and t represents the time in seconds.
When will the ball hit the ground?
SOLUTION:
Substitute h = 0 in the equation and solve for t.
Since t cannot be negative, t = 2.62. Therefore, it takes about 2.62 seconds to hit the ground.
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. Then graph the function.
59.
SOLUTION: The equation is already in vertex form.
The vertex is at (1, 5). The line of symmetry is x = 1.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
60.
SOLUTION:
The vertex is at (–3, –26). The line of symmetry is x = –3.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
61.
SOLUTION:
The vertex is at (–2, 14). The line of symmetry is x = –2.
Here, a, the coefficient of x2, is less than zero.
So, the graph opens down.
62.
SOLUTION:
The vertex is at (–6, –83). The line of symmetry is x = –6.
Here, a, the coefficient of x2, is greater than zero.
So, the graph opens up.
63. The graph at the right shows a product of 2 numbers with a sum of 10. Find a function that models this product and use it to determine the two numbers that would give a maximum product.
SOLUTION:
The standard form of a quadratic function is .
The graph intersects the x-axis at 0 and 10. So:
c = 0 and 100a + 10b = 0. The vertex is at (5, 25). Therefore: 25a + 5b = 25 Solve the equations.
b = 10 and a = –1. Substitute the values of a, b, and c in the function.
The function f (x) is maximum when x = 5. So, one number should be 5. The sum of the numbers is 10. Therefore, the other number is 5.
Graph each quadratic inequality.
64.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (–2, 2).
Shade the region that contains (–2, 2).
65.
SOLUTION:
Graph the related function .
Since the inequality symbol is <, the parabola should be dashed. Test the point (2, –4).
Shade the region that contains the point (2, –4).
66.
SOLUTION:
Graph the related function .
Since the inequality symbol is >, the parabola should be dashed Test the point (3, 2).
Shade the region that contains the point (3, 2).
67.
SOLUTION:
Graph the related function .
Since the inequality symbol is , the parabola should be solid. Test the point (0, 0).
Shade the region that does not contain the point (0, 0). The graph of the quadratic inequality is:
68. Solomon wants to put a deck along two sides of his garden. The deck width will be the same on both sides and the total area of the garden and deck cannot exceed 500 square feet. How wide can the deck be?
SOLUTION:
The area of a rectangle is given by , where l is the length and w is the width. So:
Solve the inequality.
Since x should be positive, the solution set of the inequality is .
Therefore, the width of the deck should be between 0 and 5 feet.
Solve each inequality using a graph or algebraically.
69.
SOLUTION:
Find the roots of the related equation.
Test one value of x less than –6, one between –2 and –6, and one greater than –2. Test –7.
Test –4.
Test 0.
Therefore, the solution set is
70.
SOLUTION:
The inequality is true for all x. So, the solution is the set of real numbers.
71.
SOLUTION:
Solve the related equation .
Test one value of x less than –4, one between –4 and , and one greater than .
Test –5.
Test the point 0.
Test the point 3.
Therefore, the solution set is .
72.
SOLUTION:
Solve the related equation .
Test one value of x less than –1.69, one between –1.69 and 0.44, and one greater than 0.44. Test the point –2.
Test the point 0.
Test the point 1.
Therefore, the solution set is .
73.
SOLUTION:
Solve the related equation .
Test one value of x less than , one between and 2, and one greater than 2.
Test 0.
Test the point 1.
Test the point 3.
Therefore, the solution set of the inequality is .
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Study Guide and Review - Chapter 4