algebra 3 summer math 2019 - mhs.dcsdschools.orgmhs.dcsdschools.org/userfiles/servers/server... ·...

Name_____________________________________________________ Date_________________________

Algebra 3 Summer Mathfor students entering Algebra 3 in the Fall or Spring

Directions: Do all problems in this packet. This prep work should be completed, to the best of your ability, bythe first day of school. Do not rush through the packet, give yourself time to process and remember the skills.You will not turn in your Summer Math at registration, but you will need to bring it to class completed onthe first day of school. Please note that all of the material in the prep work was covered in previous mathclasses; there are no excuses. This prep work will be reviewed during the first week of class. The test on thisprep work will be given at the end of the first week of class and will count as your first TEST grade. Below aresome websites that contain videos or notes on skills that you may need to review.1. khanacademy.org2. youtube.com3. coolmath.com4. purplemath.com

SHOW WORKBox or Circle your final answer.

State the following formulas.1) slope

2) midpoint

3) distance

4) slope -intercept form

5) point-slope form

6) quadratic formula

7) Factor a2 - b2.

8) Factor a3 - b3.

9) Factor a3 + b3.

10) Factor a2 - 2ab + b2.

11) Factor a2 + 2ab + b2.

12) State the first 13 perfect square numbers. (Hint: 1, 4, ...)

13) State the first 6 perfect cubes. (Hint: 1, 8, ...)

1


14) Complete the following for the transformation equation f(x) = a{b(x - h)] + k.

Simplify the algebraic expression.15) -5(2x - 7) - 4x + 10

Evaluate the exponential expression.16) 5-3

Identify the intercepts.

17)

x-10 -5 5 10

y10

5

-5

-10

x-10 -5 5 10

y10

5

-5

-10

Approximate the number using a calculator. Round your answer to three decimal places.18) 6 -3.2

19) 3.3!

Solve the problem.20) The rabbit population in a forest area grows at the rate of 9% monthly. If there are 240 rabbits in September,

find how many rabbits (rounded to the nearest whole number) should be expected by next September. Usey = 240(2.7)0.09 t.

21) Your math class is going to test new digital clinometers by measuring the angle of elevation of a kite you willfly. The kite flies to an angle of 72.2° on 500 feet of string. Assuming the the string is taut, how high is thekite to the nearest foot?

500 ft

72.2°

2


Simplify the exponential expression.

22) (x5)-6

23) (-5x4y-5)(3x-1y)

24) 2x3

y2

-5

25) (10b)0

26) -22x6y5

11x12y-2

3

Write the equation of a sine function that has the given characteristics.27) The graph of y = x, shifted 9 units to the right

Use the product rule to simplify the expression.28) 48

Add or subtract terms whenever possible.29) 2 2x - 6 2x

30) 6 7 + 2 28

31) 4 + 108 + 81 + 75

Evaluate the radical expressions or indicate that the root is not a real number.

32)327

Find the degree of the polynomial.33) -2x + 9x7 + 2x6 - 17

Solve the quadratic equation using the quadratic formula.34) x2 + 10x + 50 = 0

Find the slope then describe what it means in terms of the rate of change of the dependent variable per unit change inthe independent variable.

35) The linear function f(x) = -8.5x + 35 models the percentage of people, f(x), who eat at fast food restaurantseach week x years after 1998.

3


Perform the indicated operations. Write the resulting polynomial in standard form.36) (7x8 + 2x7 + 9x6 + 1) - (2x8 + 6x7 - 2x6 - 3)

37) (5x2 + 4x + 7) + (2x2 + 5x + 9) - (5x + 2)

Find the product.38) (x - 5)2

39) (p - q)(p2 + pq + q2)

40) (1 + x3)(1 - x3)

41) (5x + 3)(8x - 1)

Use Pascal's Triangle to expand the binomial and express the result in simplified form.

42) (x3 + 5y)4

43) (x - 5 )5

44) (x + 3y)3

Solve the problem.45) For a culture of 80,000 bacteria of a certain strain, the number of bacteria N that will survive x hours is

modeled by the formula N = 8000 100 - x. After how many hours will 24,000 bacteria survive?

Factor the trinomial, or state that the trinomial is prime.46) x2 - 8x + 12

47) x2 - x - 72

48) 3x2 + 14x - 5

49) x2 - 7x + 6

50) 10x2 - 21x + 9

Solve the linear equation.51) -6x - 7 + 3(x + 1) = 5x - 5

4


Solve the right triangle for all missing sides and angles to the nearest tenth.52)

c = 23, A = 73°

Factor using the formula for the sum or difference of two cubes.53) x3 + 125

Factor completely, or state that the polynomial is prime.54) x3 - 6x2 - 25x + 150

55) 5x3 - 5

56) 8x2 - 8x - 48

Divide using long division.

57) -15x3 + 26x2 - 13x + 21-5x + 2

Solve the equation by factoring.58) x2 = x + 12

59) 9x2 - 2x = 0

Find all numbers that must be excluded from the domain of the rational expression.

60) x - 8x2 + 2x - 48

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.61) f(x) = -5x4 + 3x3 - 2x2 - 5x + 4

62) f(x) = 2x3 + 4x2 + 2x - 2

5


Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rationalexpression.

63) x2 + 10x + 24x2 + 14x + 48

64) 3x2 - 12x + 12x - 2

Multiply or divide as indicated.

65) 27x - 275

÷ 9x - 955

66) x2 + 9x + 20x2 + x - 20

∙ x2 - 25x2 - x - 20

Solve the equation by factoring.67) 8x2 + 22x + 15 = 0

Add or subtract as indicated.

68) x + 6x2 - 2x - 8

+ 5x - 1x2 - 10x + 24

69) x - 8x - 3

- 2x - 1x - 3

Find the inverse of the one-to-one function.

70) f(x) = 7x - 32

Solve the linear equation.

71) 4x3

- x = x27

- 79

Factor the difference of two squares.72) x2 - 144

Give the domain and range of the relation.73) {(5, -3), (-4, -2), (-4, 0), (0 , 2), (12, 4)}

6


Solve the formula for the specified variable.

74) F = 95C + 32 for C

75) S = 2!rh + 2!r2 for h

Find the exact values of the indicated trigonometric functions. Write fractions in lowest terms.76) Find sin B and tan B.

48

1450

Factor the perfect square trinomial.77) x2 - 12x + 36

Divide using synthetic division.

78) x5 + x3 + 2x + 3

79) (x4 + 81) ÷ (x - 3)

Solve.80) 5x + 7 + 1 = 3

81) x2 + 1 = 10

Solve the quadratic equation by completing the square.82) x2 + 16x + 47 = 0

Compute the discriminant. Then determine the number and type of solutions for the given equation.83) x2 - 5x + 6 = 0

84) 7x2 = -5x - 8

Solve the quadratic equation using the quadratic formula.85) 7x2 + 12x + 1 = 0

7


Find the distance between the pair of points.86) (7, 6) and (-6, -5)

Use the given conditions to write an equation for the line in the indicated form.87) Passing through (5 , -4) and parallel to the line whose equation is y = -7x + 6;

slope -intercept form

Graph the line whose equation is given.

88) y = - 12x + 1

x-6 -4 -2 2 4 6

y6

4

2

-2

-4

-6

x-6 -4 -2 2 4 6

y6

4

2

-2

-4

-6

89) Find the length of the legs of an isooceles right triangle if the hypotenuse is 16 cm. Give the exact answer.

Factor by grouping. Assume any variable exponents represent whole numbers.90) 5x3 + 25x2 + 9x + 45

Solve the rational equation.

91) 3y + 2

- 6y - 2

= 15y2 - 4

Use the Pythagorean theorem to find the unknown side of the right triangle.

92)

6 134

8


93)

5

3

Solve.

94)3x + 2 - 4 = 0

Determine whether the equation defines y as a function of x.95) y2 = 4x

Evaluate the function at the given value of the independent variable and simplify.96) f(x) = 3x2 - 4x + 6; f(x - 1)

97) f(x) = x3 + 6x2 - 2

; f(-2)

Solve the problem.98) The function P(x) = 0.5 x - 52 models the relationship between the number of pretzels x that a certain vendor

sells and the profit the vendor makes. Find P(800), the profit the vendor makes from selling 800 pretzels.

Evaluate the exponential expression.99) 4-3 ∙ 4

First, write the value or values of the variable that make a denominator zero. Then solve the equation.

100) 4x + 2

+ 3x - 2

= 12(x + 2)(x - 2)

Solve the problem.101) The function f(x) = 400(0.5)x/90 models the amount in pounds of a particular radioactive material stored in

a concrete vault, where x is the number of years since the material was put into the vault. Find the amount ofradioactive material in the vault after 120 years. Round to the nearest whole number.

Suppose the point (2, 4) is on the graph of y = f(x). Find a point on the graph of the given function.102) The reflection of the graph of y = f(x) across the x-axis

103) y = 3f(x)

9


Find the midpoint of the line segment whose end points are given.104) (9, 9) and (7, 4)

105) Find the exact value of cos θ. Rationalize the denominator if necessary.

9

8

Express the interval in set-builder notation and graph the interval on a number line.106) (-7, ∞)

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

107) [-2, ∞)

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

108) [-4, 2)

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

Solve the absolute value inequality. Other than ∅, use interval notation to express the solution set and graph thesolution set on a number line.

109) 4y + 123

< 4

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

Solve the radical equation, and check all proposed solutions.110) x - 5 = 3x - 5

Solve the problem.111) A certain store has a fax machine available for use by its customers. The store charges $2.10 to send the first

page and $0.50 for each subsequent page. Use an inequality to find the number of pages that can be faxed for$5.10 .

10


112) A spinner has five regions numbered 1 through 5. If the spinner is spun 100 times, we would expect about20 of the outcomes to be Region 1. It can be determined that the spinner is unbalanced if x, the number of

outcomes that result in Region 1, satisfies x - 204

≥ 1.645. Describe the number of outcomes that determine

an unbalanced spinner that is spun 100 times.

Use the graph to determine the function's domain and range. Use interval notation.

113)

x-4 -2 2 4

y4

2

-2

-4

x-4 -2 2 4

y4

2

-2

-4

114)

x-4 -2 2 4

y4

2

-2

-4

x-4 -2 2 4

y4

2

-2

-4

Find the domain of the function. Use interval notation.115) f(x) = x2 + 8

116) f(x) = 1x + 4

Solve the system of equations by the substitution method.117) y = 4x - 2

4y - 12x = 8

Solve the system by the elimination method.118) 3x - 7y = 1

-7x + 4y = -27

11


For the given functions f and g , find the indicated composition.119) f(x) = 19x2 - 3x, g(x) = 17x - 6

(f∘g)(6)

120) f(x) = -6x + 5, g(x) = 4x + 9(g∘f)(x)

Solve the quadratic equation by the square root property.121) 5x2 = 55

Use the given conditions to write an equation for the line in slope-intercept form.122) Slope = 3, passing through (7, 2)

Use a calculator to find the function value to four decimal places.123) tan 58.58 °

Solve the quadratic equation using the quadratic formula.124) 3x2 = -5x - 3

Add or subtract as indicated and write the result in standard form.125) 3i - (-3 - i)

126) (4 + 8i) + (-7 - 3i) + (2 + 3i)

127) Find the length of the leg. If your answer is not an integer, leave it in simplest radical form.

Divide and express the result in standard form.

128) 7 + 2i5 - 2i

12


129) Find the value of x and y. No decimals

Find all numbers that must be excluded from the domain of the rational expression.

130) x - 6x2 - 64

131) 9x + 3

Solve the problem.132) A square playground has a white line drawn through its diagonal. This creates two triangles whose angle

measures are 45°, 45°, and 90°. If the sides of the playground are 71 feet, use special right triangles to find thelength, x, of the diagonal.

133) What is the length of the diagonal of a square with side lengths of 7 2? Give the exact value.

Solve the problem.134) A 16-foot ladder is leaning against a house with the base of the ladder 6 feet from the house. How high up

the house does the ladder reach? If necessary, round to the nearest tenth foot.

135) Find the supplement of an angle whose measure is 51°.

136) Find the complement of an angle whose measure is 44°.

13


137) Find the exact value of sin θ. Rationalize the denominator if necessary.

6

5

Solve the absolute value equation.138) 6x + 8 + 5 = 7

Solve the quadratic equation by the square root property.139) (2x - 1)2 = 49

Write the equation that results in the desired transformation.140) The graph of y = x3, vertically compressed by a factor of 0.4

141) Write the phrase used to find the three basic trigonometric

Find the product and write the result in standard form.142) (2 - 9i)(-4 - 2i)

143) What is the length of an altitude of an equilateral triangle with side lengths 8 3? Give the exact value.

Use a calculator to find the function value to four decimal places.144) cos 52.1°

145) What is the angle of elevation of the sun when a 35-foot flag pole casts a 22-foot shadow? Round to thenearest tenth of a degree.

Solve the quadratic equation using the quadratic formula.146) x2 + 4x = 9

Find the acute angle θ, to the nearest hundredth of a degree, for the given function value.147) tan θ = 0.6775

148) sin θ = 0.674

149) cos θ = 0.4765

14


Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to thegraph of f.

150) f(x) = x2, g(x) = x2 + 3

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

Simplify the complex rational expression.

151)

1x + 2

3x2 - 4

152) The time, in seconds, that it takes an object to fall a distance d, in feet, is given by the algebraic expressiond16. Find how long it will take a ball dropped from the top of a building 83 feet tall to hit the ground.

Write the answer in simplified radical form.

Determine whether the relation is a function.153) {(-4, 2), (-3, 2), (4, 7), (6 , -6)}

Solve the linear inequality. Other than ∅, use interval notation to express the solution set and graph the solution set ona number line.

154) -3(2x + 1) ≥ 2[3x - 2(x + 3 )]

15

Answer KeyTestname: ALGEBRA 3_SUMMER MATH 2019

1) m = y2 - y1x2 - x1

2) Mx1+ x22

, y1+ y22

3) d = x2 - x1 2 + y2 - y1 2

4) y = mx + b5) y - y1 = m x - x1

6) x = -b ± b2 - 4ac2a

7) ( a + b) (a - b)8) (a - b) (a2 + ab + b2 )9) (a + b) (a2 - ab + b2 )10) (a - b)2

11) (a + b)212) {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169}13) {1, 8, 27, 64, 125, 216}14) a. translate

b. dilatec. reflects; x-axisd. reflects; y-axise. right; leftf. up; downg. stretches; compressesh. compresses; stretches

15) -14x + 45

16) 1125

17) (-2, 0), (0, 8)18) 0.00319) 42.55620) 70221) 476 ft

22) 1x30

23) -15x3

y4

24) y10

32x15

25) 1

26) -8y21

x18

27) y = x - 928) 4 329) -4 2x30) 10 731) 11 3 + 1132) 333) degree 734) {-5 ± 5i}35) m = -8.5; the percentage of people eating at fast food

restaurants each week has decreased at a rate of-8.5% per year after 1998.

36) 5x8 - 4x7 + 11x6 + 437) 7x2 + 4x + 1438) x2 - 10x + 2539) p3 - q3

40) 1 - x6

41) 40x2 + 19x - 342) x12 + 20x9y + 150x6y2 + 500x3y3 + 625y4

43) x5 - 25x4 + 250x3 - 1250x2 + 3125x - 312544) x3 + 9x2y + 27xy2 + 27y345) 91 hr46) (x - 2)(x - 6)47) (x + 8)(x - 9)48) (3x - 1)(x + 5)49) (x - 1)(x - 6)50) (2x - 3)(5x - 3)

51) 18

52) B = 17°, a = 22, b = 6.753) (x + 5)(x2 - 5x + 25)54) (x - 6)(x + 5)(x - 5)55) 5(x - 1)(x2 + x + 1)56) 8(x + 2)(x - 3)

57) 3x2 - 4x + 1 + 19-5x + 2

58) {-3, 4}

59) 0, 29

60) x ≠ -8, x ≠ 661) falls to the left and falls to the right62) falls to the left and rises to the right

63) x + 4x + 8

, x ≠ -8, -6

64) 3x - 6, x ≠ 2

1


65) 33

66) x + 5x - 4

67) - 54, - 32

68) 6x2 + 9x - 38(x - 4)(x + 2)(x - 6)

69) - x + 7x - 3

70) f-1(x) = 2x + 37

71) - 218

72) (x + 12)(x - 12)73) domain: {5 , 0, -4, 12}; range: {-3, -2, 0, 2, 4}

74) C = 59(F - 32)

75) h = S - 2!r22!r

76) sin B = 725; tan B = 7

24

77) (x - 6)2

78) x4 - 3x3 + 10x2 - 30x + 90 + -268x + 3

79) x3 + 3x2 + 9x + 27 + 162x - 3

80) - 35

81) 3 11, - 3 1182) {-8 - 17, -8 + 17}83) 1; two unequal real solutions84) -199; no real solution

85) -6 - 297

, -6 + 297

86) 29087) y = - 7x + 31

88)

x-6-5-4-3-2-1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

x-6-5-4-3-2-1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

89) 8 290) (x + 5)(5x2 + 9)91) {-11}92) 7 293) 2294) 6295) y is not a function of x96) 3x2 - 10x + 1397) - 198) $348

99) 116

100) -2, 2; ∅101) 159 pounds102) (2, -4)103) (2, 12)

104) (8, 132)

105) 179

106) {x x > -7}

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

107) {x x ≥ -2}

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

108) {x -4 ≤ x < 2}

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

109) (-6, 0)

-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9

110) {10}

2


111) 7 pages or fewer112) Fewer than 14 or more than 26 outcomes113) domain: [0, ∞) or x ≥ 0

range: [2, ∞) or y ≥ -1114) domain: (-∞, ∞) OR all reals

range: (-∞, 3] or y < 2115) (-∞, ∞) or all reals116) (-∞, -4) ∪ (-4, ∞) or x < -1 or x > -1117) {(4, 14)}118) {(5, 2)}119) 174,816120) -24x + 29121) {- 11, 11}122) y = 3x - 19123) 1.6370

124) - 56

± i 116

125) 3 + 4i126) -1 + 8i127) x = 17 3; y = 34

128) 3129

+ 2429 i

129) x = 11 2; y = 11 + 11 3130) x ≠ 8, x ≠ -8131) x ≠ -3132) 100.4 ft133) 14134) 14.8 ft135) 129°136) 46°

137) 6 6161

138) - 53 , - 1

139) {-3, 4}140) y = 0.4x3141)142) -26 + 32i143) 12144) 0.6143145) 57.8 °146) {-2 - 13, -2 + 13}147) 34.12°148) 42.38 °149) 61.54°

150) g shifts the graph of f vertically up 3 units

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

151) x - 23

152) 834

seconds

153) Function

154) -∞, 98

-3 -2 -1 0 1 2 3 4 5-3 -2 -1 0 1 2 3 4 5

3

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