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Fort Zumwalt School District

PRE-AP CALCULUS

SUMMER REVIEW PACKET

Name: ______________________________________________________________________

1. You are expected to work on this packet throughout the summer to keep your skills fresh.

2. This packet is to be handed in to your Pre AP Calculus teacher on the first day of the school

year.

3. All work must be shown in the packet OR on separate paper attached to the packet. 4. Completion of this packet will be counted toward your first quarter grade.

5. An assessment of these skills WILL occur the first week of school.

These are the skills you should be proficient in performing before you get to Pre-AP

Calculus.

Linear Equations – graphing lines, defining slope, writing linear equations in point-slope and slope-

intercept forms

Factoring – factor a variety of polynomial expressions, factor out GCF, difference of squares,

perfect square trinomials, general trinomials, factoring by grouping

Solve Quadratic Equations by Factoring- example: 2𝑥2 + 14𝑥 + 24 = 0

Solving Quadratic Equations using the Quadratic Formula

Graphing: parabolas, square root function, absolute value, exponential functions

Properties of Logarithms

Solving Systems of Linear Equations

Properties of Exponents

Right Triangle Trig and Special Right Triangles

Equations of Circles and Graphing Circles

Rational Expressions

Kahnacademy.org is a great resource for tutorials and extra practice.

2

x

y

x

Geometry:

Find the value of x.

1. 2. 3.

9 5

12 5 12

4. A square has perimeter 12 cm. Find the length of the diagonal.

Solve for x and y.

5. 6.

45°

x 4

x

45°

y

Pythagorean Theorem (right triangles): a 2 b2 c 2

x x

8

2

4

y

7. 8.

x

The lengths of the legs of a right triangle are given. Find the hypotenuse.

9. a = 3, b = 4 10. a = 4 , b = 6

Find the distance between the points P1 and P2.

11. P1 = (3, 6); P2 = (-4, -2)

Solve the problem.

12. Find the length of each side of the triangle determined by the three points P1, P2, and P3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both.

P1 = (-5, -4), P2 = (-3, 4), P3 = (0, -1)

13. Find the midpoint of the line segment joining the points P1 and P2.

P1 = (5, -8); P2 = (-1, -6)

Solve the problem.

14. If (-2, -1) is the endpoint of a line segment, and (-1, 2) is its midpoint, find the other endpoint.

y

x

3

15. Find the area A of a triangle with height 8 in and base 7 in.

16. Find the area A and circumference C of a circle of radius 6 in. Express the answer in terms of π.

17. Find the area of the shaded region. Express the answer in terms of π. The sides of the square measure 9 cm and a circle is inscribed.

18. A diagonal of a rectangle measures 12 cm. The length of the rectangle is 7 cm. Find the area and perimeter of the rectangle. Round to two decimal places, if necessary.

Solve. If necessary, round to the nearest tenth.

19. If a tree 22.5 feet tall casts a shadow that is 9 feet long, find the height of a tree casting a shadow

that is 17 feet long.

20. Solve the right triangle using the information given. Round answers to two decimal places, if

necessary.

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Simplify each of the following.

21. 22. √(2𝑥8) 23. √−643

24.

25. √11

9 26. (√5 − √6)(√5 + √2)

Rationalize

27. 1

√2 28.

3

2−√5

5

Complex Numbers:

Simplify.

29. √−49 30. 6√−12 31 . 6(2 8i) 3(5 7i)

32. (3 − 4𝑖)2 33. (6 4i)(6 4i)

Rationalize

34. 1+6𝑖

5𝑖

For of x num r -

r a s r s r nd s r r

A w ys s subs ons nd

-1

o s fy for rfor ng ny o r on

p 5 1 5 P

i 5 M subs on

s w

S fy

Subs

ny o r r w n +, -, , or ( w ys s fy

)

p 2i(3 i) 2(3i) 2i(i) D s r

6i 2i 2

S fy

6i 2( 1) M subs on

2 6i S fy nd r wr n x for

S , no nsw r n n ‘ ’ n or

RA I NA I

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Equations of Lines:

35. State the slope and y-intercept of the linear equation: 5x – 4y = 8.

36. Find the x-intercept and y-intercept of the equation: 2x – y = 5

37. Write the equation in standard form: y = 7x – 5

Write the equation of the line in slope-intercept form with the following conditions:

38. slope = -5 and passes through the point (-3, -8)

39. passes through the points (4, 3) and (7, -2)

40. x-intercept = 3 and y-intercept = 2

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Graphing:

Graph each function, inequality, and/or system.

41. 3x 4 y 12

42 2x y 4

x y 2

43. y 4x 2

44. 𝑦 + 2 = |𝑥 + 1|

45. 𝑦 > 2|𝑥| − 1 46. 𝑦 + 4 = (𝑥 − 1)2

Vertex:

x-intercept(s):

y-intercept(s):

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Systems of Equations:

Solve each system of equations. Use any method.

2x y 4 47.

3x 2 y 1

2x y 4 48 .

3x y 14

2w 5z 13 49.

6w 3z 10

6

2 2 4

Subst tut on:

S 1 on for 1 r .

R rr . substitue o 2

on.

m nat on

F nd oppos ff s for 1 r .

M y on(s) by ons (s). Add ons r ( os 1 r ).

S for r r . S for r .

n nsw r k o n or on o s for 2

r . y 6 3x s 1

on for y

2 2(6- 3 ) 4

8x 16

x 2 solve

2

on

s r

s fy

6x 2 y 12 y 1

on by 2

2 - 2 4 ff s of y r oppos

8x 16 dd

x 2 s fy

Substitute x 2 k o or

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Exponennts

Express each of the following in simplest form. Answers should not have any negative exponents.

50. 5ao 51. 3𝑐

𝑐−1 52. 2𝑒𝑓−1

𝑒−1 53. (𝑛3𝑝−1)

2

(𝑛𝑝)−2

54. (3𝑚)2(5𝑚7) 55. (2𝑎−3)2 56. (−𝑏3𝑐4)5 57. 4𝑚(3𝑎−2)

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Polynomials

Simplify.

58. 3x3 9 7x

2 x

3 59. 7m 6 (2m 5) 60. 3(𝑏 − 4𝑎2) − 2(−3𝑎2 − 7𝑏)

Multiply.

61. (3a + 1)(a – 2) 62. (2x + 3)(5x – 6)

63. (2c – 5)2

64. (5x + 7y)(5x – 7y)

Factor completely.

65. z

2 4z 12 66. 6 5x x

2 67. 2k

2 2k – 60

68. 10b4 15b

2 69. 9c

2 30c 25 70. 9n

2 4

71. 27z3 8 72. 2mn 2mt 2sn 2st 73. 12𝑥2 − 52𝑥 − 40

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Factor completely (continued)

74. 15𝑥2 + 14𝑥 − 8 75. 8𝑥2 − 18𝑥𝑦 + 9𝑦2 76. 16𝑥2 − 56𝑥 + 49

77. (2x + 1)2 + 3(2x + 1) – 4 78. 3(x + 5)2(2x - 1)2 + 4(x + 5)3(2x - 1)

Solve each equation.

79. x

2 4x 12 0 80. x

2 25 10x 81. 3x2 + 7x - 20 = 0 82. x(x - 10) + 24 = 0

Solve the equation by the Square Root Method.

83. (x - 5)2 = 4 84. (x + 6)2 = 10

s quad at quat ons, t y to f t f st and s t h f t qua to o. S f you ab . If th quad at d s N t , us quad at f mu a.

EX: x

4 x 21

x

4 x 21 0

(x 3)(x 7) 0

x 3 0

x

x 7 0

x 7

Set equal to zero FIRST .

Now factor.

Set each factor equal to zero.

Solve each for x.

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Solve the equation by completing the square or any other method.

85. x2 + 4x = 3 86. x2 + 5x - 5 = 0

QUADRATIC FORMULA – allows you to solve any quadratic for all its real and imaginary roots.

EX: Solve the equation: x2 2x 3 0 a= 1, b = 2, c= 3 𝑥 = −𝑏±√𝑏2−4𝑎𝑐

2𝑎

𝑥 = −2±√22−4(1)(3)

2(1) 𝑥 =

−2±√−8

2 𝑥 =

−2±2𝑖√2

2 x = -1 ± i√2

Find the real solutions, if any, of the equation. Use the quadratic formula.

87. x2 + 4x - 9 = 0 88. 8x2 - x + 4 = 0

Find the real solutions of the equation.

89. x4 – 10x2 + 9 = 0 90. 4(x + 1)2 + 7(x + 1) + 3 = 0

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Find the quotient and the remainder using long division.

91. 6x8 - 10x3 divided by 2x 92. x2 + 3x - 45 divided by x + 9

Use synthetic division to find the quotient and the remainder.

93. x2 + 6x + 4 is divided by x + 2 94. x3 - x2 + 5 is divided by x + 2

95. -3x3 - 16x2 + 17x + 30 is divided by x + 6 96. x5 + 8x4 + 17x3 + 12x2 + 12x + 13 is divided by x + 5

Use synthetic division to determine whether x - c is a factor of the given polynomial.

97. 2x3 - 11x2 - 35x + 98; x + 2

Use synthetic division to determine whether x - c is a factor of the given polynomial.

98. x3 - 5x2 - 29x + 105; x + 5 99. 3x3 - 20x2 - 21x + 98; x – 7

2𝑥2 − 3𝑥 + 3 +1

𝑥 + 3

14

Use synthetic division to find the quotient and the remainder.

100. 6x5 - 5x4 + x - 4 is divided by x + ½

Algebra Review- Functions

Find the value for the function.

101. Find f(4) when f(x) = x2 + 5x - 1. 102. Find f(-9) when f(x) = |x|- 6.

103. Find -f(x) when f(x) = -3x2 + 5x + 1.

Solve the problem.

104. If f(x) = 8x3 + 7x2 - x + C and f(-2) = 1, what is the value of C?

105. Find f(-x) when f(x) = -2x2 + 5x - 4.

Find the domain of the function.

106. f(x) = 𝑥

𝑥2+5 107. 𝑔(𝑥) =

𝑥−3

𝑥3−49𝑥 108. ℎ(𝑥) =

𝑥

√𝑥−4

For the given functions f and g, find the requested composite function.

109. f(x) = 6x + 7, g(x) = 2x - 1; find (f o g)(x). 110. 𝑓(𝑥) = 7

𝑥−5 , 𝑔(𝑥) =

4

3𝑥; find (f o g)(x)

For the given functions f and g, find the requested composite function value.

111. f(x) = x + 3, g(x) = 3x; find (𝑓𝑜𝑔) (4). 112. f(x) = 9x2 - 6x , g(x) = 9x - 9; find (f o g)(2).

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Find the inverse, 𝒇−𝟏(x), if possible.

113. f (x) 5x 2 114. 𝑓(𝑥) = 1

2𝑥 −

1

3 115. 𝑓(𝑥) =

−8𝑥+6

−9𝑥−6

ALGEBRA REVIEW (RATIONAL EXPRESSIONS) Reduce the rational expression to lowest terms.

116. 𝑥2−36

𝑥−6 117.

2𝑥+2

6𝑥2+16𝑥+10 118.

7𝑥2−60𝑥+32

𝑥−8 119.

(𝑥2+2)∙8−(8𝑥+9)∙3𝑥

(𝑥2+2)3

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Perform the indicated operations and simplify the result. Leave the answer in factored form.

120. 121. 2𝑥−2

𝑥∙

2𝑥2

9𝑥−9 122. 123.

𝑥2+7𝑥+12

𝑥2+12𝑥+27

𝑥2+4𝑥

𝑥2+11𝑥+18

124. 125. 126. 1

(𝑥−4)(𝑥−5)−

3

(𝑥−5)(𝑥−2) 127.

2

𝑥2−3𝑥+2+

6

𝑥2−1

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128. 1+

1

𝑧

𝑧+1 129. 130.

1

𝑥+3+

1

𝑥−31

𝑥2−9

131.

8

9−𝑥+

9

𝑥−9

𝑥2−9

Solving Rational Equations

132. 133. 134. 9

𝑥+5−

7

𝑥−5=

14

(𝑥+5)(𝑥−5) 135.

𝑥

5=

𝑥

𝑥−5− 1

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Change the following to logarithmic form.

136. 28 = 256 137. 4−3 = 1

64 138. 𝑘5 = 𝑦 139. (√3)𝑥= 81

Simplify

140. log4 16 141. log8 1 142. log51

125 143. log 1000

ALGEBRA REVIEW (Circles)

144. Write the equation of the circle if r=8 and the center is (5, 9).

145. Write the equation of the circle with radius = √6 and center is (-3, 4).

146. Name the center and the radius of the circle with the given equation.

a. (𝑥 + 2)2 + (𝑦 − 1)2 = 4 b. 𝑥2 + (𝑦 + 8)2 = 81

147. Sketch the graph of the circle 𝑥2 + (𝑦 − 2)2 = 20

The standard form of an equation of a circle is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 where the center of the circle is (h, k) and the radius is r.