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Pre-Calculus: Cumulative Review

Name: ___________________________

Classify the function. Write the transformations and find the domain and range of the transformed

graph.

1. 𝑦 = 2(−𝑥 + 11)2 − 4 2. 𝑦 =1

2√2𝑥 − 183

3. Write the equation of a cubic graph that has been reflected over the y-axis and shifted left 5 units.

4. Write the equation of a square root graph that has been vertically stretched 5, moved up 6 units,

and right 2 units.

Given that 𝒇(𝒙) = 𝒙𝟐 + 𝟏, 𝒈(𝒙) = 𝟐𝒙 + 𝟓, and 𝒉(𝒙) =𝟑

𝒙+𝟐 find each of the following.

5. 𝑓(𝑔(𝑥)) 6. (𝑓 + ℎ)(𝑥) 7. 𝑓(𝑥)

𝑔(𝑥)

Graph the following piecewise functions.

8. 9.

𝑓(𝑥) = {

−𝑥 − 4 𝑖𝑓 𝑥 < −2

−1

2𝑥 𝑖𝑓 − 2 ≤ 𝑥 ≤ 2

−1 𝑖𝑓 𝑥 > 2

𝑓(𝑥) = {

𝑥 + 4 𝑖𝑓 − 6 ≤ 𝑥 < 2−6 𝑖𝑓 𝑥 = 2

−𝑥 + 2 𝑖𝑓 𝑥 > 2

2

Answer the following using the piecewise function from problem 9.

10. Find the domain of the function. 11. Find the range of the function.

12. Find 𝑓(−8) 13. Find 𝑓(0) 14. Find 𝑓(11)

Use the rational function 𝒇(𝒙) =−𝟐𝒙𝟐+𝟐𝒙+𝟐𝟒

𝒙𝟐+𝟑𝒙 to find each of the following.

15. Graph the function in the space below.

16. x-intercept(s) 17. y-intercept 18. Hole

19. VA 20. HA 21. OA

22. HA-intercepts 23. local max 24. local min

25. increasing intervals 26. decreasing intervals 27. domain

28. Range 29. all applicable limits

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Find the inverse for each of the following.

30. 𝑦 = 3𝑥2 + 4 31. 𝑓(𝑥) = log4(𝑥 + 1) − 6 32. 𝑦 =𝑥+8

2𝑥

Solve the following equations. Show all of your steps for full credit. If necessary, round to the nearest

tenth.

33. log2𝑥 + log2(𝑥 + 7) = 3 34. 2 ln(𝑥 − 5) − ln 3 = ln 9

35. (1

5)

−3𝑏−1∙ 252𝑏 = 53 36. 42𝑏+9 = 3−5𝑏+7 37. 𝑒8𝑛−1 = 6

38. The number, N, of bacteria present in a culture at time t, in hours, is represented by the function

𝑵(𝒕) = 𝟏𝟑𝟎𝟎𝒆𝟎.𝟎𝟏𝒕. After how many hours will the population be 3000?

39. The half-life of Cobalt 60 is 4,945 years. If an object currently has 40 grams of Colbalt 60, how

long will it take for there to be only 9 grams remaining? Use the formula 𝑨 = 𝑷 (𝟏

𝟐)

𝒕𝒉⁄

4

Graph each of the following. Find the transformations, amplitude, period, domain, and range.

40. 𝑦 = −2 sin(4𝜃 + 𝜋) + 1

41. 𝑦 = 3 cos1

3(𝜃 −

𝜋

2) − 2

42. You go to the carnival and decide to ride the Ferris Wheel. The wheel is 5 feet off of the ground

and has a diameter of 38 feet. The wheel makes a complete revolution every 12 seconds. Draw a

graph and write a cosine function to model the height of the Ferris Wheel after 𝒕 seconds. Use your

equation to determine the height after 8 seconds.

For each of the following scenarios find all missing sides and all missing angles. You should draw a

picture of the given information first so that you know what you have and can choose the best

solution method. One of the following problems has two triangles! Round decimal answers to the

nearest tenth.

43. 𝐵 = 42°, 𝐶 = 38°, 𝑎 = 11 𝑐𝑚 44. 𝐶 = 90°, 𝑎 = 14.1 𝑚, 𝑏 = 4 𝑚

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45. 𝑎 = 7 𝑦𝑑, 𝑏 = 11 𝑦𝑑, 𝑐 = 16 𝑦𝑑 46. 𝐴 = 108°, 𝐵 = 10°, 𝑎 = 16 𝑖𝑛

47. 𝐴 = 64°, 𝑎 = 26 𝑐𝑚, 𝑐 = 27 𝑐𝑚 48. 𝐵 = 114°, 𝑎 = 27 𝑚, 𝑐 = 23 𝑚

49. Find the area of the triangle in #45 with Heron’s formula.

50. Find the area of the triangle in #44 using the information given in the problem and a variation of

the formula 𝑨𝒓𝒆𝒂 =𝟏

𝟐(𝒃)(𝒄)(𝐬𝐢𝐧 𝑨)

Find ALL exact solutions for the following equations.

51. cos 𝜃 = −√3

2 52. 2 sin 𝜃 − 1 = 0 53. 3 tan 𝜃 + 3 = 0

54. cos(3𝜃) =1

2 55. 2sin (5𝜃 +

𝜋

3) + √2 = 0

Find the exact solutions of the following on the interval 𝟎 ≤ 𝜽 < 𝟐𝝅.

56. cos 𝜃 = −√3

2 57. 2 sin 𝜃 − 1 = 0 58. 3 tan 𝜃 + 3 = 0

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59. cos(3𝜃) =1

2 60. 2sin (5𝜃 +

𝜋

3) + √2 = 0

Use a calculator to solve each equation on the interval [𝟎, 𝟐𝝅).

61. sin 𝜃 = −0.89 62. cos 𝜃 = 0.23 63. tan 𝜃 = −2.15

Prove that the following statements are true.

64. sin 𝜃(cot 𝜃 + tan 𝜃) = sec 𝜃 65. sec4 𝜃 − sec2 𝜃 = tan4 𝜃 + tan2 𝜃

66. cos(𝛼−𝛽)

cos 𝛼 cos 𝛽= 1 + tan 𝛼 tan 𝛽 67. sec 𝜃 − tan 𝜃 =

cos 𝜃

1+sin 𝜃

Classify the following conic sections. Then write the conic section in standard form.

68. 𝑥2 + 𝑦2 + 8𝑥 + 11 = 0 69. 25𝑥2 + 16𝑦2 + 150𝑥 − 32𝑦 − 159 = 0

7

Given the conic section, find the required information.

70. (𝑦 − 6)2 = 8(𝑥 + 4)

vertex:

direction of opening:

focus:

directrix:

endpoints of the latus rectum:

length of the latus rectum:

71. (𝑥 − 6)2 + (𝑦 + 2)2 = 10

center:

radius:

72. 𝑥2

16+

(𝑦−4)2

9= 1

center:

vertices:

co-vertices:

foci:

major axis:

minor axis:

8

73. 𝑥2

16−

(𝑦−4)2

9= 1

center:

vertices:

foci:

asymptotes:

Use the given information to find an equation for each of the following.

74. An ellipse with the vertices (−6, 15) and (−6, −3) and the foci (−6, 6 + 3√5)and

(−6, 6 − 3√5).

75. A hyperbola with the center (−8, 4), a vertex at (3, 4), and a focus at (−8 + √146, 4).

A vector v had an initial point P and a terminal point Q. Find its position vector and write it in both

component and 𝒂𝒊 + 𝒃𝒋 form.

76. 𝑃 = (−11, 4) and 𝑄 = (3, −8) 77. 𝑃 = (0, 13) and 𝑄 = (7, −1)

Use the vectors to answer the following questions.

𝒗 = −𝟑𝒊 + 𝟖𝒋 𝒘 = 𝟔𝒊 − 𝟒𝒋 𝒛 = 𝟕𝒋

78. 4𝑣 − 𝑤 79. 3𝑤 + 7𝑧

80. 𝑣 ∙ 𝑤 81. 𝑣 ∙ 𝑧

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82. Find the unit vector for v.

83. Miss Rice is flying an airplane. The airplane has an airspeed of 350 kilometers per hour in the

direction 𝑺𝟐𝟎°𝑾. The wind velocity is 42 kilometers per hour in the direction 𝑺𝟏𝟎°𝑬. Find the

resultant vector representing the path of the plane relative to the ground. What is the ground speed

of the plane? What is its direction?

84. Miss Rice hits a softball at a bearing of 𝟐𝟑°, with a speed of 17 m/sec. There are strong gusty

winds outside, blowing 23 m/sec with a bearing of 𝟑𝟑𝟔°. What is the softball’s true speed and

direction?

Eliminate the parameter of the following parametric equations. Write the resulting equation in

standard form.

85. 𝑥 = 4 cos 𝑡 , 𝑦 = 5 sin 𝑡 for 0 ≤ 𝑡 ≤ 2𝜋 86. 𝑥 = 𝑡 + 2, 𝑦 = 𝑡2 − 7 for 0 ≤ 𝑡 ≤ 4

Plot each polar coordinate on the graph.

87. 𝐴(5, 225°)

88. 𝐵(−2, 𝜋)

89. 𝐶 (4,5𝜋

3)

90. 𝐷(−1, 90°)

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91. Convert #87 and #89 into rectangular form. Answers should be exact.

Convert each of the following rectangular coordinates to polar coordinates. Answers for 𝜽 should be

in radians, exact answers if possible. If you need to have decimals, please round to the nearest

thousandth.

92. (−5.3, 2.1) 93. (2, 2√3)

Find the other 3 polar coordinates that represent the same point for −𝟐𝝅 ≤ 𝜽 ≤ 𝟐𝝅.

94. (6,7𝜋

4)

Convert the following polar equations into rectangular equations.

95. 𝑟 = 2 96. 𝜃 =𝜋

3

97. 𝑟 = −4

cos 𝜃 98. 𝑟 = 2 csc 𝜃

99. 𝑟 = 2 cos 𝜃 − 8 sin 𝜃 100. 𝑟 =3

2−4 sin 𝜃

Convert each of the following rectangular equations to polar form.

101. 𝑥2 + 𝑦2 = 6𝑥 102. 𝑦 = −√3

3𝑥

Determine the type of sequence (arithmetic or geometric), find the explicit formula, the recursive

formula, the 5th term, and the sum of the first five terms.

103. 4

3, 1,

3

4,

9

16, … 104. 6.5, 4.4, 2.3, 0.2, …

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105. Find the explicit formula and recursive formula for the arithmetic sequence where the 10th term is

−121 and the 32nd term is −319.

Given the recursive sequence, find the first five terms.

106. 𝑎𝑛 = 3 + 𝑎𝑛−1 107. 𝑎𝑛 = 1 − 𝑛 + 𝑎𝑛−1 + 𝑎𝑛−2

𝑎1 = 3 𝑎1 = −2

𝑎2 = 7

Find the sum of the sequence.

108. 109.

Write the sequence in summation notation.

110. 4 + 7 + 10 + 13 + ⋯ + [3(33) + 1] 111. 1 +2

7+

4

49+

8

343+ ⋯ +

128

823543

112. Miss Rice’s Honors Pre-Calc students are becoming increasingly addicted to math each day. On

the first day they do 7 math problems. On the second, they do 15. On the third, they do 23, and so on.

How many math problems do they do on the 35th day? How many math problems have they done

total?

∑(3𝑘 − 𝑘−1)

7

𝑘=2

∑ (2 (2

3)

𝑘−1

)

∞

𝑘=1

12

113.

114.

Find the limit of each of the following by hand. Remember, the “limit” at a specific x-value is what the

y-value is supposed to be, not what it actually is.

115. lim𝑥→−3

𝑓(𝑥) where 𝑓(𝑥) = {−𝑥2 − 10𝑥 − 24 𝑖𝑓 𝑥 ≤ −3

2𝑥 + 3 𝑖𝑓 𝑥 > −3

13

116. lim𝑥→−1−

𝑓(𝑥) where 𝑓(𝑥) = {𝑥 𝑖𝑓 𝑥 < −1

−𝑥2 + 2𝑥 𝑖𝑓 𝑥 ≥ −1

117. lim𝑥→−2+

𝑓(𝑥) were 𝑓(𝑥) = {𝑥2 𝑖𝑓 𝑥 ≤ −2

−𝑥

2+ 3 𝑖𝑓 𝑥 > −2

118. lim𝑥→−3

𝑥−3

𝑥2+5𝑥+4 119. lim

𝑥→1(−𝑥3 + 2𝑥2 + 1)

120. lim𝑥→−9

7 121. lim𝑥→5

𝑥2−25

𝑥+5

122. lim𝑥→−2

𝑥2−2𝑥−8

𝑥+2 123. lim

𝑥→3

𝑥2−7𝑥+12

𝑥−3