Algebra 1B - Chapter 10 - Part 3 Mathlete:

Date Assigned

Topic Homework (due the next day)

J

Fri 2/3

10.6 Quadratic Formula Chapter 10 Lesson 6 parts 1 & 2

In packet: Using the Quadratic Formula 1-17odd

Mon 2/6

10.7 The Discriminant Chapter 10 Lesson 7

In textbook: 10.7 Page 594-595; 1-3, 4-36 even

Tue 2/7

Chapter 10 Lesson 1 Exploring Quadratic Graphs Chapter 10 Lesson 2 Quadratic Functions

In textbook: 10.2 Page 560-561; 1-10. For 11-13,18,20,22,28: graph axis of symmetry, vertex, y-intercepts, x-intercepts and corresponding points for

Wed 2/8

Chapter 10 Lesson 3 Solving Quadratic Equations Chapter 10 Lesson 5 Completing the Square

Do the remaining problems in lessons 3 and 5.

Thu 2/9

Chapter 10 Story Problems In textbook: Page 555; 38,44 Page 561; 35,37 Page 567-569; 19-21,42 Page 574; 21,23

Fri 2/10

Chapter 10 Review In packet: Chapter 10 Test Review

Mon 2/13

Chapter 10 Test

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Chapter 10 Lesson 6 Using the Quadratic Formula

Warm Up Find the value of c to complete the square for each expression. 1. x2 + 6x + x 2. x' + 7x + c 3. x' — 9x + c

Solve each equation by completing the square. 4. x' — 10x + 24 = 0 5. x' + 16x — 36 = 0

6. 3x2 + 12x — 15 = 0 7. 2x2 — 2x — 112 = 0

Objectives — To use the quadratic formula when solving quadratic equations

The is found by completing the square of the general quadratic equation.

Example 1 — Using the Quadratic Formula Use the quadratic formula to solve the equation x2 — 4x =117

Example 2— Finding Approximate Solutions Use the quadratic formula to solve the equation —3x2 +5x —2 = 0. Round to the nearest hundredth.

Chapter 10 Lesson 6 Part 2 Using the Quadratic Formula

Objectives — To use the quadratic formula when solving quadratic equations To choose an appropriate method for solving a quadratic equation

Example 3 — Real—World Problem Solving — Using the Vertical Motion Formula In real—world situations, you must decide whether a solution makes sense. (Negative values)

A child throws a ball upward with an initial upward velocity of 15 ft/s from a height of 2 ft. If no one catches the ball, how long will it be in the air? Round to the nearest hundredth of a second.

Step 1 Use the vertical motion formula 2

h = —16t + vt + c where v = initial upward velocity and c = starting height

Step 2 Use the quadratic formula

Example 4— Choosing an Appropriate Method

Method When to Use

Graphing Use if you have a graphing calculator handy.

Square Roots Use if the equation has no x term.

Factoring Use if you can factor the equation easily.

Completing the Square Use if the x2 term is 1, but you cannot factor the equation easily.

Quadratic Formula Use if the equation cannot be factored easily or at all.

Which method(s) would you choose to solve each equation? Justify your reasoning.

2 a. 5x + 8x — 14 = 0 b. 25x

2 - 169 = 0 c. x

2— 2x— 3 = 0 d. x

2— 5x + 3 = 0

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Kuta Software - Infinite Algebra 1 Name

Using the Quadratic Formula Date Period

Solve each equation with the quadratic formula.

1) m2 - 5m - 14 = 0 2) b2 - 4b + 4 = 0

3) 2m2 + 2m - 12 = 0 4) 2x2 - 3x - 5 = 0

5 x2 +4x+3 = 0

6) 2x2 + 3x - 20 = 0

7) 4b2 +8b+7 = 4 8) 2m2 - 7m - 13 = -10

-1-

9) 2x2 - 3x -15 = 5 10) x2 + 2x - 1 = 2

11) 2k2 + 9k= -7 12) 5r2 = 80

13) 2x2 - 36 = x 14) 5x2 + 9x = -4

15) k2 - 31 - 2k = -6 - 3k2 - 2k

16) 9n2 = 4 + 7n

17) 8n2 + 4n - 16 = -n2

18) 8n2 + 7n - 15 = -7

-2-

Chapter 10 Lesson 7 Using the Discriminant

Evaluate b2 — 4ac for the given values of a, b, and c. 1. a = 3,b = 4, c = 8 2. a = —2,b = 0, c = 9

3. a = 11,b = —5, c = 7

4. 3x2 — 7x + 1 = 0

5. 4x2 + x — / = 6. x2 — 12x + 35 = 0

Objectives — To find the number of solutions of a quadratic equation

You can tell how many solutions a quadratic equation has you solve it

by using the

The is the expression under the radical in the quadratic formula.

X —b+Vb 4ac

2a

Property of the Discriminant

For the quadratic equation ca2 + bx + c = 0, where a 0, you can use the value of the discriminant to determine the number of solutions.

If b2 — 4ac > 0, then there are two solutions. If b2 — 4ac = 0, then there is one solution. If b2 — 4ac < 0, then there are no solutions.

Example 1 — Using the Discriminant Find the number of solutions for each equation. a. 2 2x — 3 b. 3x2 — 4x = 7 c. 5x2 + 8 = 2x

Example 2 — Real—World Problem Solving — Using the Discriminant

A construction worker tosses an apple to a fellow worker who is 20 ft above the ground. The starting height of the apple is 5 ft. Its initial velocity is 25 ft/s. Will the apple reach the second worker?

2 h = —16t + vt + c where v = initial upward velocity and c = starting height

Ii

Chapter 10 Lesson 1 Exploring Quadratic graphs

Warm Up 1. Graph the equations y = x2 and y = 3x2 on the same coordinate plane.

2. A. Describe how the graphs are alike.

B. Describe how the graphs are different

V

3. Predict how the graph of y = 1x2 will be similar to and different from the graph of y = 3

1 4. Graph y = -- x2 . Were your predictions correct? Explain.

3

V Objectives — To graph quadratic functions of the form y = ax2

To graph quadratic functions of the faun y = ax2 + c

Standard Foini A quadratic function is a function that can be written in the form y = ax2 + bx + c. Vocabulary

If a> 0 in y = ax2 +bx+c,

then the parabola opens

and the vertex is the point

or the point.

If a< 0 in y = ax2 +bx+c,

then the parabola opens

and the vertex is the point.

or the point

O The graph of a quadratic function is a U-shaped curve called a

O The fold or line that divides the parabola into two matching halves is

called the

• The highest or lowest point of a parabola is called the

Example 1 — Identifying a vertex

Identify the vertex of the graph. Tell whether it is a minimum or maximum. a. b.

5 4. 3

-1

Example 4— Graphing y = ax2 +c

Graph the quadratic functions y = 2x2 , I G -

y = 2x2 + 3 and y = 2x2 — 2 . Compare the graphs.

5 4 3 2

-2 -II

Describe how positive and negative values

1

of c change the graph.

7- .

12 - I.

Chapter 10 Lesson 2 Quadratic Functions

Warm Up

Evaluate the expression --b

for the following values of a and b. 2a

1. a = —6, b — 4 2. a = 15, b = 20

3. a — —8, b — —56 4. a = —9, b = 108

Graph each function. (Pick 5 points, make a table and graph.)

5. y = x2 6. y = —x2 +2

A

Objectives — To graph quadratic functions of the form y = ax2 +bx+ c To graph quadratic inequalities

Property — Graph of a Quadratic Function

The graph of y = ax2 + bx + c , where a 0, has the line x = -

-b as its

2a

The x-coordinate of the

is —b

. 2a

Example 1 — Graphing y = ax2 + bx + c

Graph y = x2 — 6x + 9 Step 1 Find the axis of symmetry and the coordinates of the vertex

Step 2 Find two other points on the graph

Use the y-intercept

Choose another value for x on the same side of the vertex as the y-intercept.

Step 3 Reflect the two points across the axis of symmetry. Then draw the parabola.

Example 2— Real World Problem Solving

A ball is thrown into the air with as initial upward velocity of 48 ft/s. Its height h in feet after t seconds is given by the function h = —16t 2 + 48f + 4.

a. In how many seconds will the ball reach its maximum height?

b. What is the ball's maximum height?

9 49

100 4

Chapter 10 Lesson 3 Solving Quadratic Equations

Warm Up Simplify each expression.

4.V[44

2. —V-8-T: 3. +V121

5. V0.25 6. +VT.H.

1. Find the x intercepts of each graph (by graphing a. y = 2x — 3 b. y = x2 + 3x — 4

5

-4

'3

• 2

3

2.a. Solve 2x — 3 = 0

b. Is the solution of 2x — 3 = 0 the same as the x-intercept of y = 2x — 3?

3. Do the x-intercepts that you found in Question lb satisfy the equation x2 + 3x — 4 = 0?

4.a. Graph y = + x — 6

-2

-2

4

-5

-6

b. Find the x-intercepts of the graph of y = x2 + x — 6

c. Do the values you found in part (b) satisfy the equation x2 + x — 6 = 0?

Objectives — To solve quadratic equations by graphing To solve quadratic equations using square roots

Definition — Standard Form of a Quadratic Function — ax 2 + bx + c = 0

The solutions of a quadratic equation and the related x-intercepts are often called

Or

Example 2 — Using Square Roots Solve each equation a. t2 — 25 = 0 b. 3n2 + 12 = 12 c. 292 + 32 = 0

Example 3— Real World Problem Solving A circular fountain has a depth of 3 feet and a volume of 1800 ft3. Find the fountain's radius.

Chapter 10 Lesson 5 Completing the Square

Warm Up Find each square.

1. (d - 4)2 2. (x + 11)2 3. (k - 8)2

Factor each expression. 4. b 2 + 10b + 25 5. t 2 + 14t + 49 6. n2 - 18n + 81

Objectives - To solve quadratic equations by completing the square

In a perfect square trinomial, with a = 1, c must be the square of half of b.

b By adding (

2-2) to x 2 bx, you change the expression in to a perfect square trinomial.

This process is called

Example 1 - Finding n to Complete the Square a. Find the value of n such that x2 - 12x +n is a perfect square trinomial.

b. Find the value of n such that x2 + 22x +n is a perfect square trinomial.

Example 2— Solving x2 + bx = c a. Solve the equation x2 +9x =136

Step 1 Write the left side as a perfect square.

Step 2 Solve the equation

b. Solve the equation m2 —6m = 247 Step 1 Write the left side as a perfect square.

Step 2 Solve the equation

Algebra 1 Chapter 10 Test Review

Name:

Hour:

Show all work and circle your answers to receive credit.

1. Graph the function. Label the axis of symmetry, the vertex, and two points on each side of the vertex. y=-2x2 —x+3

2. Use the Zero-Product Property to solve the equation: (x + 3)(5x + 2) = 0

3. Order the group of quadratic functions from widest to narrowest graph. 1

y=--2

x2, y=-2x2,y=x2,y=--x

2

3 2

4. James is planning to expand his square bedroom. If each side of his original room is increased by 5 feet, the new total area of the room will be 196 square feet. Find the length of each side of the original room.

5. Simplify each expression.

a. (—V49)2 b. c. —N/719 d. —(V49)2 c. V-49

A

6. Determine how many solutions each of the following equations has.

a. 4x2 —3x +2 = 0 b. —4x2 +2x+5 = 0 c. 4x2 —8x +4 = 0

7. Solve the equation using the Quadratic Formula: 5x2 — 13x — 2 = 0.

8. Determine the number of real roots of 3x2 + 13x + 5 = 0.

9. Solve the equation by factoring: 2b2 — 4b = 6.

10. Solve x2 + 10x — 5 = 0 by Completing the Square.

11. A ball is thrown into the air with an upward velocity of 30 ft/s. Its height h in feet after t seconds is given

by the function h = —16t2 +30t +12. Find the time at which the ball reaches its maximum height. Find the maximum height.

Use the graph below to answer questions 12-14.

12. What is the equation of the axis of symmetry? What is the vertex? Is the vertex a minimum or maximum?

13. Is the discriminant of the equation corresponding to the graph positive, negative, or zero?

14. What are the solution(s) of the equation corresponding to the graph?

15. Find the vertex of the graph of the function y = 3x2 + 7x-8 . Then determine if the graph would open upward or downward.

16. Graph f>2x 2 +3x —1.

F'

›

17. Solve 3x +5 = 32 by taking square roots.

18. Find the value of n such that x2 —15x + n is a perfect square trinomial.

19. Find the number of solutions for the equation x2 +9 = 0.

20. Draw a graph that satisfies each of the following conditions. A

a. The leading coefficient is positive and the discriminant is negative.

b. The leading coefficient is negative and the discriminant is zero.

A

c. There is no linear tetrn and the discriminant is positive.

Solve the quadratic equation by any legal method. If the equation has no solution, write no solution.

. 4x2 — 25 = 0 22. 2x2 +5x —12 = 0

Solve the equation by completing the square. YOU MUST SHOW ALL OF YOUR WORK TO BE ELIGIBLE FOR FULL CREDIT

23. x2 +6x-7 = 0 24. 5x2 +12x = 2

Use the quadratic formula to solve the equation. YOU MUST SHOW ALL OF YOUR WORK TO BE ELIGIBLE FOR FULL CREDIT

25. 3x2 —2x + 7 = 0 26. 8x2 +5x = —7

27. Find the side of a square with an area of 116 ft2 .

28. For the equation —3x2 + 2x + k = 0, find all the values of k such that the equation has two solutions. Show work to support your answer.

29. The formula h = —16t2 + vt models the height of a model rocket, where h is the height in meters, t is the time in seconds, and v is the initial vertical velocity in meters per second. If the model rocket is fired at an initial vertical velocity of 62 meters per second, will the rocket ever reach a height of 60 meters? Show work to support your answer.

30. Graph the quadratic function. Label the vertex, x-intercepts and y-intercept. y=x2 -6x+8

vertex:

x-intercepts:

y-intercept:

A