SCHECK
HILLEL COMMUNITY
SCHOOL
Students Entering
Precalculus Honors
&
AP Precalculus
2023
Summer Assignment
Dear Student,
Welcome to Precalculus Honors and AP Precalculus. These courses are designed to prepare you for AP Calculus. You
will master higher algebraic manipulations, function analysis, and trigonometry. Every topic will be interpreted verbally,
numerically, analytically, and graphically, and then analyzed via these lenses with increasing layers of depth and rigor.
You will apply conceptual understanding of problems and use your mathematical reasoning to solve real-world
problems. You are responsible for completing this packet before the start of classes in August.
The work in this packet has been designed to reinforce the skills you have previously learned. It is strongly
recommended that you complete a portion of this packet each week of your summer vacation. Do not leave it all for the
end. In this way, the work will not overwhelm you.
Helpful Websites:
KhanAcademy.org
VirtualNerd.com (Algebra 2)
PurpleMath.com
Name: ________________________ Class: ___________________ Date: __________
ID: A
Honors
PrecalculusSummer
Summer Packet
AP
Pre-Calculus
Packet
Please show all work in the spaces provided.
The answers are provided at the end of the packet.
Algebraic Manipulation
1. Evaluate
3x − y
if x = 5 and y = −3
3(x − y)
6. Subtract (−3g 2 + 2g − 9) from (g 2 − 4g − 6)
.
.
2. Multiply (−x)(−3y)(−5z)
7. Multiply (6n − 3)(5n − 7)
Ê 2y ˆ Ê y ˆ Ê y + 3 ˆ˜
˜˜
3. Simplify: ÁÁÁ x ˜˜˜ ÁÁÁ 2x ˜˜˜ ÁÁÁ x
Ë ¯Ë
¯Ë
¯
.
.
4.
8. Expand (3x 4 − 7y 3 ) 2
5x −3 y 2
x 5 y −1 z 0
⋅
(2xy 3 ) −2
xy
.
9. Multiply
.
5. Simplify
18xy
2
7a b
3
2
ˆÊ
ˆ
ÊÁ 2
Á x + x − 3 ˜˜ ÁÁ 3x 2 − x + 3 ˜˜
¯Ë
¯
Ë
2
÷
12x y
35a 2 b
.
.
1
Name: ________________________
ID: A
11. Factor completely
10. Factor completely
a) a 2 − 6a − 40
a) −5x 4 y 2 + 20xy 3 + 15xy 4
b) 6y 2 + 13y − 5
b) 3x 3 + x 2 − 15x − 5
c) 12m3 n − 75mn
c) 20x 2 − 125y 2
d) 49x 2 − 100y 2
d) 4x 2 − 12xy + 9y 2
e) 6xp + 42x − 5yp − 35y
e) x 2 − 6x + 9 − 4y 2
.
2
Name: ________________________
ID: A
15. Divide using synthetic division
12. Factor the expression completely.
125a 3 − 8b 6
(3x 5 + 5x 4 + x + 5) ÷ (x + 2)
.
13. Factor the expression completely.
ˆ4
ˆ5
Ê
Ê
4 ÁÁ x 2 + 5 ˜˜ (3x)(x − 1) 4 + ÁÁ x 2 + 5 ˜˜ (9)(x − 1) 3
¯
¯
Ë
Ë
.
ˆ Ê
ˆ
Ê
16. Divide: ÁÁ x 5 + x 4 − 8x 3 + x + 2 ˜˜ ÷ ÁÁ x 2 + x − 7 ˜˜
¯ Ë
¯
Ë
.
14. Factor the expression completely.
1
1
−
ÁÊÁ x 2 + 4 ˆ˜˜ 2 + 2 ÁÊÁ x 2 + 4 ˜ˆ˜ 2
¯
¯
Ë
Ë
.
3
Name: ________________________
17. Simplify
ID: A
21. Simplify.
6y + 30
2
y − 25
1
x
x
1−
y
1+
.
18. Simplify
.
a2 − x2
a
⋅
2
3x − 3a
a
22. Simplify the compound fractional expression.
5
4
−
x−1 x+1
x
1
+
x−1 x+1
.
19. Simplify.
.
(4w 2 − 3wy)(w + y)
2
23. Simplify.
2
(3y − 4w)(5w − y )
x−1
5
−
6x − 18 4x 2 − 14x + 6
.
20. Simplify
x 3 − 64
x 2 − 16
÷
x 3 + 64 x 2 − 4x + 16
.
24. Perform the subtraction and simplify.
x
1
3
−
−
x − x − 20 x + 4 x − 5
2
.
.
4
Name: ________________________
ID: A
26. Evaluate without a calculator:
25. Simplify
a) −3 3 −3 + 2 3 162 + 3
3
ÊÁ 1 ˆ˜ −4
a) ÁÁÁÁ − ˜˜˜˜
Ë 2¯
81
Ê
ˆ
b) 4 15 ÁÁÁ 4 − 3 5 ˜˜˜
Ë
¯
b) − 32
−
c) 27
Ê
c) ÁÁÁ
Ë
ˆÊ
5x ˜˜˜ ÁÁÁ
¯Ë
3+
3−
3
5
5
3
ˆ
5x ˜˜˜
¯
d) .0049
3
2
3
2
d) 2m2 ⋅ 4m ⋅ 4m−2
.
ÊÁ 3 ˆ˜
ÁÁ 2 ˜˜
Á
˜
e) ÁÁÁ 3p ˜˜˜
ÁÁ
˜˜
Á
˜
Ë
¯
−2
27. Simplify
a)
ÊÁ 1
ÁÁ
ÁÁ 2 −2
ÁÁ x y
f) ÁÁÁ
−7
ÁÁ
ÁÁ
ÁÁ yx 4
Ë
g)
x y
243a 20 b 25
ˆ˜ 4
˜˜
˜˜
˜˜
˜˜
˜˜
˜˜
˜˜
˜
¯
ÊÁ 1 ˆ˜
ÁÁ 2 ˜˜
ÁÁ
˜
ÁÁ x y 2 ˜˜˜
ÁÁ
˜˜
Á
˜
Ë
¯
2
5
b) 2
−5
4
1
2
.
5
3
18a 2 b ⋅ 3
3
12ab 5
Name: ________________________
ID: A
30. Simplify each expression and write the result in the
form a + bi.
28. Rationalize the denominator and simplify
a)
2a
b
4
b)
a) 6 − (2 + 9i) + (−1 + 4i)
3
4
4
b) (7 − i)(3 + 3i)
5
27
c) (5 − 7i)
3
c)
d)
5−
6
d)
3−i
2 + 3i
e)
−5 − 3i
−4i
x
y
2
5
.
29. Rationalize the numerator and simplify
a)
3+
4
f) i 1795
7
.
b)
x+3 −5
3x − 66
.
6
2
Name: ________________________
ID: A
Equations and Inequalities
31. Solve 3(x + 2) =
1
(12x + 4) − 5x
4
34. Solve the equation.
|3x − 5| = 7
.
.
35. Solve the equation for the indicated variable:
32. Solve the equation.
y+1
2
1
y + (y − 4) =
5
2
4
a)
ax + b
= 11 , solve for x.
cx + d
b)
x
x
= c − , solve for x.
a
b
c)
x
x
= c − , solve for b.
a
b
.
33. Solve.
4
5 23
+ =
x + 3 6 18
.
7
Name: ________________________
ID: A
37. Solve by completing the square.
36. Solve each equation:
x 2 + 8x + 22 = 0
a) n(9 − 3n)(2n + 5) = 0
b) x 2 + 8x − 20 = 0
.
38. Solve the equation by completing the square.
2x 2 − 12x − 7 = 0
c) 20x(x − 1) = 42 − 9x
d) −8x 2 + 46x − 30 = 0
.
39. Solve by using the Quadratic Formula.
x 2 − 6x = −10
e)
3x 2 − 6x = 10
.
8
Name: ________________________
ID: A
43. Solve
a)
x+2 +4= 7
40. Find all real solutions of the equation.
x 6 − 9x 3 − 10 = 0
.
b) 2 =
41. Find all solutions of the equation and express them
in the form a + bi.
25x 2 + 16 = 0
.
.
42. Solve
1
3
2(6x − 3) − 4 = 0
.
9
3b − 2 −
10 − b
Name: ________________________
ID: A
46. Solve the nonlinear inequality. Express the solution
using interval notation and graph the solution set.
44. Solve the inequality. Graph the solution on a
number line.
|3x + 9| ≥ 6
x 2 + 3x > 10
.
.
47. Solve the nonlinear inequality. Express the solution
using interval notation and graph the solution set.
45. Solve and graph your answer on a real number
line: |x − 3 | − 2 < 6
x
1
x+1
−
≤
3 x−2
4
.
10
Name: ________________________
ID: A
Lines and Coordinate Geometry
48. Graph the line of each equation:
49. Find the slope of:
a) x + 3y = 6
a) a line passing through (-4, 4) and (2, -5)
b) a line parallel to y = 2x + 7
c) the line whose equation is 2x − 3y = 12
b) y = −2x + 4
d) the line whose equation is y = −5
e) a line perpendicular to 6x + 5y = 9
.
50. Determine whether the lines are parallel,
perpendicular, or neither. Explain your reasoning.
c) x = 3
2x + 3y = 12
3x + 2y = 24
11
Name: ________________________
ID: A
53. Solve the system of equations by graphing.
51. Find an equation, in point-slope form, of the line
that satisfies the given conditions.
2x + 3y = −27
11x − 3y = −12
Through (–1, –11); perpendicular to the line
passing through (3, 1) and (7, –1).
Then, rewrite the equation in standard form.
.
52. Solve the system of equations by substitution
8x − 2y = 10
3x − y = 9
.
54. Solve the system of equations by elimination
2x − 3y = 6
9y − 6x = 9
12
Name: ________________________
ID: A
57. Find the solution set for the following system of
inequalities.
55. Solve they system of equations by any method.
0.6x + 1.6y = 2.8
x + 2y ≤ −4
−0.05x + 0.08y = −0.02
3x − 2y > −4
.
56. Solve the system of equations by any method.
2
5
1
x+ y=
3
6
4
58. A pair of points is graphed.
1
1
−1
x−
y=
5
10
10
(a) Find the distance between them.
.
(b) Find the midpoint of the segment that joins
them.
13
Name: ________________________
ID: A
60. Find the center and radius of the circle.
59. Prove whether or not the points A(4, 3), B(2, 6),
C(7, 5), and D(5, 8) form a square.
x 2 + y 2 − 6x + 4y − 3 = 0
(a) The center is
(b) The radius is
(c) Graph and label important points
(d) Domain:
Range:
.
.
14
Name: ________________________
ID: A
Functions
62. The graph of a function g is given.
61. If f(x) = −2x 2 + x + 3 , evaluate each of the
following:
a) f(−2)
b) f(3m)
(a) Find g(–5). __________________
c) f(p 5 )
(b) Find g(–3). __________________
(c) Find g(–1). __________________
(d) Find g(1). __________________
(e) Find g(3). __________________
(f) Find the domain of g. __________________
d) f(x + h)
(g) Find the range of g. __________________
(h) Estimate the values of x for which is g(x) = −1
.
15
Name: ________________________
ID: A
63. Consider the following function:
f(x) = 3x 2 − 12x + 7
64. Consider the following function:
f(x) = −2x 2 − 12x − 14
a) Write the quadratic function in vertex form.
a) Write the quadratic function in vertex form.
b) State the vertex and whether the graph has a
minimum or maximum.
b) State the vertex and whether the graph has a
minimum or maximum.
c) State the equation of the axis of symmetry.
c) State the equation of the axis of symmetry.
d) Find the x intercepts and approximate the
values.
d) Find the x intercepts and approximate the
values.
e) Find the y intercept
e) Find the y intercept
f) Graph using all the points.
f) Graph using all the points.
16
Name: ________________________
ID: A
68. Sketch the graph of the function using a table of
values (hint: find the domain first).
65. Find the domain of the function.
h(x) =
8x − 7
f(x) = − x + 2 + 3
.
66. Find the domain of the function.
f(x) =
x+9
x2 − 4
State the domain and range.
.
.
67. Find the domain of the function.
g(x) =
x 2 − 4x − 32
17
Name: ________________________
ID: A
69. Evaluate the piecewise defined function at the
indicated values.
ÔÏÔÔ 2
ÔÔÔ x + 4x
ÔÔ
f(x) = ÌÔ x
ÔÔ
ÔÔ −9
ÔÔÓ
if
x ≤ −3
if
−3< x ≤ 1
if
x>1
70. Sketch the graph of the piecewise defined function.
ÏÔÔ
if x < −1
ÔÔ 1
ÔÔ
ÔÔ 2
f(x) = ÔÌ x
if − 1 ≤ x ≤ 1
ÔÔ
ÔÔ −x + 3 if x > 1
ÔÔÓ
(a) Evaluate f(−4) .
ÁÊ 7 ˜ˆ
(b) Evaluate f ÁÁÁÁ − ˜˜˜˜ .
Ë 2¯
(c) Evaluate f(−3) .
.
(d) Evaluate f(0) .
(e) Evaluate f(35) .
.
18
Name: ________________________
71. f(x) =
x−1
2x + 1
g(x) = x 5
ID: A
and
72. Find the inverse of each function.
h(x) = 3x − 5
a) f(x) =
Given the functions f, g, and h above, find the
following:
3x − 7
12 − 5x
a) ÊÁË g û f ˆ˜¯ (x )
b) ÊÁË f û h ˆ˜¯ (x )
b) g(x) = 3x 2 + 24x − 1
.
c) ÊÁË h û f û g ˆ˜¯ (x )
.
19
Name: ________________________
ID: A
Problem Solving
You must use an ALGEBRAIC process for all of the word problems.
Correct answers obtained by guess-and-check will not obtain full credit.
73. Find four consecutive odd integers whose sum is
464.
75. An orange has 15 calories more than a
grapefruit. Twenty oranges and ten grapefruits
have 1800 calories together. Jim ate three
oranges and half a grapefruit. How many
calories did Jim eat?
.
.
76. A sports club publishes a monthly newsletter.
Expenses are $0.90 for printing and mailing a copy,
plus $600 total for research and writing. Write a
function that represents the total monthly cost of
publishing the newsletter and use it to determine
the cost of sending 1000 copies.
74. The product of 3 and a number, decreased by 8,
is the same as twice the number, increased by
15. Find the number.
.
.
20
Name: ________________________
ID: A
79. Find the value of x. Then, find the perimeter of
77. A company manufactures and sells small weather
radios. If the cost of producting the radio can be
expressed as C(x) = 10,000 + 30x and the revenue
produced from the sales can be expressed as
R(x) = 50x , how many radios must be produced and
sold for the company to make a profit? (If you
don’t know the meaning of a term, look it up!)
the triangle.
.
80.
Find the value of x in each problem.
a)
.
78. Five times the supplement of an angle is 630° more
than its complement. Find the angle, its
supplement, and its complement.
b)
.
.
21
Name: ________________________
ID: A
81. The area of a triangle is 72 in2 and the base is 8
83. Solve the right triangle.
in. Find the height.
(a) Find b. Please give the answer to two decimal
places.
.
82. In the figure below, find a polynomial expression
that represents the area of the shaded region.
(b) Find r. Please give the answer to two decimal
places.
(c) Find m∠A.
.
.
22
Name: ________________________
ID: A
84. Phyllis invested $13,000, a portion earning a
1
simple interest rate of 4 % per year and the rest
2
earning a rate of 4% per year. After one year the
total interest earned on these investments was
$555. How much money did she invest at each
rate?
86. A cylindrical can has a volume of 90π cm3 and is
10 cm tall.
a) What is its diameter?
b) If the can has a bottom, but no top, what is its
sufrace area?
1
__________ is invested at 4 %
2
__________ is invested at 4%
85. A wire 230 in. long is cut into two pieces. One
piece is formed into a square and the other into a
circle. If the two figures have the same area, what
are the lengths of the two pieces of wire (to the
nearest tenth of an inch)?
.
__________ in.
23
ID: A
Honors
Precalculus
SummerPacket
Packet
AP
Pre-Calculus
Summer
Answer Section
NUMERIC RESPONSE
1. 3 / 4
2. −15xyz
3. 2x
4.
4y + 3
5
4x 11 y 4
15y 2
5.
2bx
6. 4g 2 − 6g + 3
7. 30n 2 − 57n + 21
8. 9x 8 − 42x 4 y 3 + 49y 6
9. 3x 4 + 2x 3 − 7x 2 + 6x − 9
10. a) (a − 10) (a + 4)
b) ÁÊË 3y − 1 ˜ˆ¯ ÁÊË 2y + 5 ˜ˆ¯
c) 3mn (2m − 5) (2m + 5)
d) ÁÊË 7x − 10y ˜ˆ¯ ÁÊË 7x + 10y ˜ˆ¯
e) ÁÊË 6x − 5y ˜ˆ¯ ÁÊË p + 7 ˜ˆ¯
ˆ
ˆ
Ê
Ê
11. a) 5x ⋅ y 2 ⋅ ÁÁ −x 3 + 4y + 3y 2 ˜˜
b) (3x + 1) ⋅ ÁÁ x 2 − 5 ˜˜
c) 5(2x − 5y) ⋅ (2x + 5y)
¯
¯
Ë
Ë
2
d) ÊÁË 2x − 3y ˆ˜¯
e) ÊÁË x − 3 − 2y ˆ˜¯ ÊÁË x − 3 + 2y ˆ˜¯
ˆ Ê
ˆ
Ê
12. ÁÁ 5a − 2b 2 ˜˜ ⋅ ÁÁ 25a 2 + 10a ⋅ b 2 + 4b 4 ˜˜
¯ Ë
¯
Ë
4
ˆ
ˆ
Ê
Ê
13. 3 ÁÁ x 2 + 5 ˜˜ ⋅ (x − 1) 3 ⋅ ÁÁ 7x 2 − 4x + 15 ˜˜
¯
¯
Ë
Ë
14.
x2 + 6
x2 + 4
15. 3x 4 − x 3 + 2x 2 − 4x + 9 −
16. x 3 − x + 1 +
17.
13
x+2
−7x + 9
x2 + x − 7
6
y−5
a+x
−3a
−w
19. Ê
5 ÁË w − y ˆ˜¯
18.
20.
x 2 + 4x + 16
2
(x + 4 )
1
ID: A
21.
y (x + 1 )
x ÊÁË y − x ˆ˜¯
x+9
x + 2x − 1
7x − 2
23.
6(2x − 1)(x − 3)
22.
24.
2
−3x − 7
(x − 5) ⋅ (x + 4)
25. a) 12
3
d) 32m
3
3 + 6
6
c) 3 − 5x
b) 16 15 − 60 3
3
2
e)
1
9p 3
f)
x9
y 12
1
g)
x
26. a) 16
b) - 8
27. a) 3a 4 b 5
28. a)
2a
29. a)
3
1
243
d)
y3
343
= 0.000343
1,000,000
b) 36ab 2
4
b2
b)
b
1
3−
−
c)
21
8
30. a) 3 − 5i
7
15
12
b)
c) −
x ⋅ 5 y3
d)
y
15 − 3 2
1
Ê
3ÁÁÁ
Ë
ˆ
x + 3 + 5 ˜˜˜
¯
b) 24 + 18i
c) −24 − 70i
d)
3 11
−
⋅i
13 13
e)
3 5
− i
4 4
f) −i
COMPLETION
31. x = −1
45
32. y =
13
33. x = 6
2
3
11d − b
35. a) x =
a − 11c
34. x = 4,x = −
36. a) n = 0, 3, or
37. x = −4 ± i
b) x =
−5
2
cab
a+b
b) x = −10, 2
c) b =
6
7
c) x = − ,x =
5
4
6
5 2
2
39. x = 3 ± i
38. x = 3 ±
40. x =
3
−ax
ax
or
x − ac
ac − x
10 ,x = −1
2
d) x = 5 or
3
4
e) x =
3±
39
3
ID: A
4
4
41. x = − ⋅ i, ⋅ i
5
5
11
6
43. a) x = 7 b) b = 6 (reject b = 1)
44. x ≤ −5 or x ≥ −1
42. x =
45. −5 < x < 11
46. (−∞,−5) ∪ (2,∞)
47. (−∞,−1] ∪ (2,6]
SHORT ANSWER
48. a)
49. a) −
b)
3
2
b) 2
c)
2
3
d) 0
e)
c)
5
6
−2
−3
, m2 =
3
2
Since these slopes are neither equal nor opposite reciprocals, the lines are neither parallel nor perpendicular.
51. y + 11 = 2(x + 1)
50. m1 =
2x − y = 9
52. (-4, -21)
3
ID: A
53. ÊÁË −3, − 7 ˜ˆ¯
54. No solution.
55. (2, 1)
ÊÁ 1 1 ˆ˜
56. ÁÁÁÁ − , ˜˜˜˜
Ë 4 2¯
57.
58. 2 10 ; (1, –2)
59. You need to find the slopes and lengths of all the sides. If the consecutive sides are all congruent and
perpendicular, then the shape is a square.
60. (3, –2); 4
ESSAY
61. a) −7 b) −18m2 + 3m + 3 c) −2p 10 + p 5 + 3
62. 3; 2; –2; 1; 0; [–5, 3]; [–2, 3]
d) −2x 2 − 4xh − 2h 2 + x + h + 3
4
ID: A
63. a) f(x) = 3(x − 2) 2 − 5
b) Vertex (2, -5) minimum
c) x=2
d) x intercepts x = 2 ±
5
3
≈ 2 ± 1.3
≈ 0.7 and 3.3
e) f(0) = 7
f)
64. a) f(x) = −2(x + 3) 2 + 4
b) Vertex (-3, 4) maximum
c) x = -3
d) x intercepts x = −3 ± 2
≈ −3 ± 1.4
≈ −1.6 and − 4.4
e) f(0) = −14
f)
ÈÍ
Í 7 ˆ˜
65. ÍÍÍÍ ,∞ ˜˜˜˜
ÍÎ 8 ¯
5
ID: A
66. (−∞,−2) ∪ (−2,2) ∪ (2,∞)
67. (−∞,−4] ∪ [8,∞)
68.
D [−2,∞)
R (−∞,3]
7
69. 0; − ; –3; 0; –9
4
70.
ÊÁ x − 1 ˆ˜ 5
˜˜
Ê
ˆ
71. a) ÁË g û f ˜¯ (x ) = ÁÁÁ
ÁË 2x − 1 ˜˜¯
x−2
b) ÁÊË f û h ˜ˆ¯ (x ) =
2x − 3
ÊÁ 5
ˆ
Á x − 1 ˜˜˜
˜˜ − 5
c) ÁÊË h û f û g ˜ˆ¯ (x ) = 3 ÁÁÁÁ 5
Á 2x + 1 ˜˜
Ë
¯
12x + 7
72. a) f − 1 (x) =
5x + 3
b) g −1 (x) =
x + 49
−4
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PROBLEM
73.
74.
75.
76.
113, 115, 117, 119
n = 23
220 calories
The cost is $1500 for 1000 copies.
6
ID: A
77. They must sell at least 500 radios.
78. angle=45°, comp=45°, supp = 135°
79. x=12
P=102
80. a) x = 8
b) x = 4 5
81. h = 18in
82. 6x + 22
83. 30.03; 41.75; 44
84. $7,000; $6,000
85. 108.1, 121.9
86. d = 6, SA= 69π
7