Pre-Calculus 2024 Summer Packet

Congratulations!! You have made it to Pre-Calculus!!! We commend you on your choice regardless of the level!
In order to be successful in Pre-Calculus, you need to do some preparatory work this summer. As you will discover, the
skills you learned in Algebra courses and Geometry will be utilized and necessary to be successful in Pre-Calculus.
Attached is a review packet that will help you to focus on the mathematical skills and content you will need to use in
solving calculus problems. These problems deal with skills and content you studied in previous math classes. Use your
notes, the internet or other resources and seek out assistance to help you remember these concepts. Take your time and
make sure you understand the concepts behind each problem.
You are responsible for completing this summer assignment before the school year begins in August. Be prepared to turn
in this assignment during the first week of school. Teachers will be available for after school tutoring over this content
the first week of school. This review packet will not necessarily be covered in class as it is all previously learned
material.
Optional tutoring will be available the week before school starts at Cy-Fair HS.

Please do all problems on a separate sheet of paper. Show all steps and all work. You should attempt to
complete every problem without a calculator. You may use a calculator as a tool to help solve the problem if
necessary and you should check your work using a calculator when possible.
#1 – 10, simplify:
1 2 3  16  8 −17 13 93
1. − 2. 5 − 3. (−4)   + (3)   4. − 5. −2
6 3 8  3 5 12 5 24

1 5  5 11  1 2 1  x 4x
6. 5 − + (2)   7. 6  −  8. y + y− y 9. 8  − 2  10. −
3 8  12 18  3 5 8  3 7

#11 – 25, factor the following:

11. a 2 − b 2 12. 14 xyz − 7 y 5 + 28 xz 6 13. 4 x 2 y 3 − 2 xy + 6 xy 4 14. 4x 2 y − y

15. 2 x 2 + 5 x − 18 16. 3 x 2 − 11x − 4 17. 2 x 2 + 20 x + 50 18. x 4 − y 4

19. 6 x 2 y 4 − 24 x 4 y 6 20. 9 x 2 − 12 x + 4 21. 6 x 2 − 13 xy − 5 y 2 22. 5 x 2 + 57 x − 36

23. x 3 − 9 x 2 + 2 x − 18 24. 20n3 + 12n 2 + 25n + 15 25. 12 xy + 20 xa − 84ay − 140a 2

#26 – 40, solve the following for x:

26. (4 x − 1)(8 x + 5) =
0 x 2 12 x + 8
27. 6= 28. x 2 + 6 =0 29. −2 x 2 + 7 =
13 x

x 4 2x 1
30. 3 x ( 3 x − 8 ) =−16 31. 2 x 3 + 7 x 2 − 4 x =
0 32. x + = −x 33. + =5
3 5 3 4
x −3 x + 4 1 16 y
34. = 35. 3(2 x − 7) + 1 = 36. 5 xy 2 − 3 xz 5 =
10 37. = 3 y3
x +1 x + 6 9 y + 4x

5x x + 6 2x + 3
38. =x 39. 4 x 2 + 15 =
2z 40. =
6 y4 z2 x x+6
#41 – 51: Expand by multiplication:

41. ( x − 3) 2 42. (− x + 8) 2 43. (3 x + 2) 2 44. (7 − x)(5 x − 1)

45. (2 x − 1)(3 x + 10) 46. (6 x − 7)(−2 x + 5) 47. ( 8 − x )( x − 8 ) 48. (8 x − 1)( x + 2) 2

49. ( x − 1)( x 2 − 2 x + 1) 50. (− x + 8)( x 3 − 2 x − 1) 51. (− x + 6)(2 x − 3)(3 x + 4)

#52 - 59 Combine and simplify to simplest radical form:

52. 168 53. 2 27 + 243 54. 3 20 + 2 25 + 45 55. 10 • 8

3 3 5 1
56. 57. 58. 3 • 59.
12 3 6 8

#60-68 Find the length of the missing side(s) using Pythagorean Theorem or Special Right Triangles.
60. 61. 5 62.
15 16
x 4
15
x
5 x

x
63. 16
64. 65. 45°
x x y
x
12
x
3√2
66. 67. 68.
y x
x x
30°
30° 6√6 √10
24 y y
# 69 – 76 Solve the following:
4 f ( x) − 1
69. If y = −3 , then 5 y 2 + 3 = 70. Given the equation = f ( x) , solve for f ( x).
3x

x 4 y + 3 , then 2 x + 1 =
71. If = 72. If (6ε − 9δ ) 2 + 10 =
235 , then 2ε − 3δ =

1 4 2 6 x + 10
73. If=z x − 2 x 2 + 1 + 1 then (2 z − 1) = 74. Given the equation = 14 , solve for 8 x − 2
2 x
75. 2Ω + 1= x − y , then 4Ω + 2 = 76. Given the equation 5( f ( x) − 3) + x − f ( x) =(10 x) f ( x) ,

solve for f ( x).

#77 – 92 Graph the following (on graph paper). Label key points, including x- & y-intercepts,
asymptotes, holes, etc. Identify Domain and Range:
 79. y =( x − 4 ) + 5
2
77. 4 x − 5 y =
6 78. 2 x − 6 = 2 x − y 80. y =9 − 6 x + x 2

( x − 2)
2
1 x−9
81. x + 5 y =
6 82. 4 x + xy =
6 83. − 2y =
0 84. =y
2 4 x+3
3 5
− ( x − 12 ) − 11
2
85. 5 y + 4 = 6 x − 86.=y 2 x − 10 87. y = 88. y =
5 3x + 9
(4 x + 1)( x − 2)
89. y = 8 90.
= y
6
+3 91. y = 92. y = x−3
x x−4 ( x + 1)( x − 2) x − 2 x − 15
2

#93 – 95 State the domain restrictions and solve for x.

x 3 12 3 3 x 2
93. = 94. + = 95. x+ =
x+2 4 x 4 2 x−2 x−2
96. Mr. Proton is writing a final exam for his Chemistry class. Each problem on the 100-point test will count the same
100
number of points. The function, y = , gives the point value, y, for each question when there are x number of
x
questions on the test. If Mr. Proton wants each question on the test to count between 2 and 3 points, what is a reasonable
number of questions he should put on the test?

#97-104 Transform each equation from exponential form to logarithmic form or vice versa, as required

97. 34 = 81. 98. 26 = 64 99. 163/2 = 64 100. 5 log5x = x

101. log 3 9 = 2. 102.log101000 = 3 103. log749 = 2 104. log50.2 = x

#105 – 113 Solve for x.

1
105. 4 x = 43 x+6 106. 64 x = 36 x+ 4 107. 52 − x =
25
108. 5 x = 10 109. 3 • 4x =
21 110. 3x = −18
3
111. log 2 64 = x 112. log 2 x = −3 113. log x 8 =
4
Use the following for problems #114 & #115.
The growth of a certain bacteria over time in days is represented by the exponential equation

y 2000 • (1.5)
x
= where y represents the number of bacteria and x represents the time in days.

114. How many bacteria are there after 6 days?

115. If there are 34,172 bacteria present, how many days have passed?
 #116 – 118 Find the length of each indicated arc. Leave your answers in terms of π.
116. 117. 118.

#119 – 121 Find the area of each shaded sector of a circle. Leave your answers in terms of π.
119. 120. 121.

#122 – 124 Solve

122. Find the area of one piece of pizza if the diameter of the pizza is 16 in and it is cut into 8 equal pieces.
123. The minute hand of a courthouse clock measures 12 ft. How far does the tip of the hand travel in 12
minutes? In one hour?
124. A wheelbarrow has a front wheel 1 ft in diameter. How far does the wheelbarrow travel in one complete
revolution of the front wheel?

#125 – 127 Graphing form of the equation of a circle is:

( x − h) + ( y − k ) r2; =
with center ( h, k ) & r
2 2
= radius .

Identify the center and radius of each circle and graph. Label 4 points on the graph.
125. 𝑥𝑥 2 + (𝑦𝑦 − 1)2 = 9 126. (𝑥𝑥 + 3)2 + (𝑦𝑦 − 4)2 = 18

127. 𝑥𝑥 2 + 𝑦𝑦 2 + 4𝑥𝑥 − 6𝑦𝑦 + 13 = 25 (hint: think completing the square)