Dear Students,
There are certain math skills that have been taught to you over the previous years that are necessary to
be successful in Honors Pre-Calculus. Even though you may understand the Pre-Calculus concepts, without
these skills you will consistently struggle to correctly solve problems next year. It can be frustrating for
students when they are doing Pre-Calculus but are “tripped up” by the Algebra. This summer packet is
intended to help you brush up and possibly relearn these topics.
On the following pages are problems from various important topics. All problems should be done on a
separate sheet of paper, in order, and all necessary work must be shown. Your work will be due the first day
of class and will be graded. You will also take short quiz over the material after we have reviewed together in
class. To make the most of this packet and start the semester off right, we recommend that you spend some
quality time with the packet this summer. Do not try to finish it before school is out for the summer- we want
the topics to be fresh in your minds for the fall! Do not attempt to do it all the night before the first day of
classes- it will be a daunting task! We also recommend that you do not rely on your calculator. Almost all
problems should be possible to solve using paper, pencil, and your brain.
Don’t fake your way through these problems. If you find yourself needing some assistance, please go to
one of the websites listed below. In some cases, these sites have full instructions on certain techniques. Do not
hesitate to use the web for your assistance. We want to emphasize that the concepts in this packet will be used
throughout the upcoming year and understanding these concepts will greatly aid in your success.
Good Luck!
Ms. Aiello & Mr. Hymes
For Algebra Topics: http://www.khanacademy.org
Honors Pre-Calculus
Summer Packet 2017
http://www.purplemath.com/modules/index.htm
Page 1 of 5
Honors Pre-Calculus Summer Packet
Topic #1 – Absolute Value
1)
Use absolute value notation to describe: The distance between
Topic #2 – Exponents
For problems #2 & #3, simplify.
 3x y
2) 
 xw 2

2 3




3
5 2
x y
3) 
 z2





x and 16 is no more than 5.

3
For problems #4 - #8, evaluate.
4) 4 x 2 for x  3
5)
 8 
6)  
 27 
1
1 / 2
81
 #3 – Simplify Radicals
Topic
For problems #9 - #10, rationalize the denominator.
3
9)
10)
7 2
2 / 3
 1 
7)  
 64 
2 / 3
1
8)
27
1 / 3
For problems #11 - #12, simplify by rationalizing the denominator.
2
4
11)
12)
3
3
2x
x
5
7 2
For problems #13 - #17 , simplify.
13)
3
16)
7 25 xy 2  4 75 xy 2  2 12 xy 2
3x  1
14)
3 3 4 x 5 y 3  7 x 3 32 x 2 y 6
15)
2 x 2 y 3 2 x  7 x 2 3 2 xy3  4 3 16 x7 y 3
17)
Topic #4 – Polynomial Multiplication and Factoring
For problems #18 - #19 , multiply.
Expand.
18)
20)
x  2x  2x 2  4
x  1  y x  1  y
19)
4 9x  2 4x 7
3  2 y 3
For problems #21 - #24 , factor completely.
21) y3  ( x  1 )3
22) 2rs  3rst  8r  12rt
24) 3xz  2 yz  6 xw  4 yw
23) 3rv  2vt  6 rs  4st
For problems #25 - #27 , factor completely. Eliminate negative exponents in the final answer.
25)
26)
27)
3x  23  9 x3x  24
6 x( 2 x  3 )4  24x2 2 x  35
2 x( 4 x  1 )2  4 x  11
Topic #5 – Operations with Rational Expressions
For problems #28 - #35 , perform the indicated operation, then simplify if possible.
28)
29)
30)
x  5x  4
2
x2  4

x2
x 2  3x  4
32)
3
9

x x1
Honors Pre-Calculus
x  4x  4 2  x

x2
3x  6
2
33)
4
7

x2 x3
4 x  16 4  x

5 x  15 2 x  6
34)
31)
x 2  2 x  63 9  x

x1
x2  x
35)
3
1

2
x  2x  1 x  1
Summer Packet 2017
3
x  x2
2

x
x  x 6
2
Page 2 of 5
Topic #6– Operations with Rational Exponents/Complex Fractions
For problems #36 - #41 , simplify.
Eliminate negative exponents and rationalize the denominator in the final answer.
36)
37)
38)
x
x
3 x  5 1 / 3 
7 x  2 1 / 3 
1 x  x / 1 x
3 x  5 2 / 3
7 x  2 2 / 3
2/3
2/3
1 x
3 x  5
7 x  2

40)
39)

3 /

x2  x2
5 x2
41)
5 x  2 
1/ 2
1
( 2 )  2 x 5 x  2 1 / 2 ( 5 )
2

5x  2

Topic #7– Solve Equations
For problems #40 - #41 , solve for
42)
 
4 3
x 7 x  1 2 / 3  7 x  11 / 3 12 x 2
3
2
x.
15x  4  4  2 x  3
43)
x 2  2x  2x  3

Topic #8– Solve Inequalities
For problems #42 - #44 , solve the inequality. State the answer in interval notation.
44)
45)
46)
4
3
2
3
2
3



x 1 x 1
x1 x2
x  2 x 1
Topic #9– Function Composition
For problems #47 - # 50, find  f  g x  using the given f ( x ) and g( x ) .
47)
f(x)
48)
1
, g( x ) 
x2
x2  4
2
f ( x )  x 2 , g( x )  x  1
49)
50)
x
f(x)
x 1
2
f(x)
, g( x )  x 3
x2
x 1
2
, g( x ) 
x2 1
Topic #10– Domain
For problems #51 - #54 , determine the domain of
51)
52)
f(x)
2
f ( x )  x , g( x )  x  4
53)
f(x)
f  g . State the domain in interval notation.
1
x2  1
, g( x )  x  3
54)
1
x
f(x)
, g( x )  x  3
1
x2  1
, g( x )  1  x
Topic #11– Inverse and Function Composition
For problems #55 - #62 , use the given f ( x ) , find f 1( x ) and state the domain of the inverse in interval notation.
55)
56)
57)
58)
f ( x )  2x  1
Honors Pre-Calculus
f ( x )  2 x 2  1 for x  0
f ( x )  3x 3  1
Summer Packet 2017
f ( x )  2 x5
Page 3 of 5
For problems #59 - #62 , find the indicated quantity using the given f ( x ) and g( x ) .
59)
60)


f ( x )  x  1 and g( x )  1  2 x Find g 1  f 1 x
61)





f ( x )  3  x and g( x )  x 3 Find f 1  g 1 x
62)

f ( x )  x  1 and g( x )  1  2 x Find f 1  g 1 2
f ( x )  3  x and g( x )  x 3 Find g 1  f 1  5
Topic #12– Quadratic Functions
IMPORTANT NOTE: OUR BOOK renames vertex and standard form.
Vertex Form (from the Alg. 2 book) is now called STANDARD FORM: f ( x ) = a(x – h)2 + k
Standard Form (from the Alg. 2 book) is now called GENERAL FORM: f ( x ) = ax2 + bx + c
For problems #63 - #64 , graph each quadratic. DO NOT use a graphing calculator. Find the line of symmetry, vertex, zero’s (if
they exist) and the y-intercept.
63)
64)
f ( x ) = –2(x – 3)2 + 7
f ( x ) = –x2 – 6x + 7
For problem #65 , convert from standard form to general form by completing the square.
65)
f ( x ) = 3x2 + 12x + 9
66)
Graph for # 66 
Find the standard form of the quadratic function shown:
For problems #67, find the standard form of the equation given the vertex and a point on the parabola. Do not use decimals.
Vertex:   2 , 1  and 3,4  is on the parabola.
67)
 3 9
For problem 68 & 69, find 2 equations in GENERAL form, one that opens up and one that opens down, whose graph have the given
x-intercepts. Do not use decimals, use integer coefficients only.
68 & 69)  3 ,0  and  2 ,0 
4

5

70) Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 120.
71) Farmer Jon has 336 feet of fencing and wants to build two identical
pens for his prize-winning pigs. The pens will be arranged as shown.
Determine the dimensions of a pen that will maximize its area.
Topic #13 – Polynomial Division
Use long division to divide. Leave any remainder as a fraction.
72) x 2  6 x  8  x  4 
73) x 4  x 2  5  x 2  2 x  1



 

Use synthetic division to divide. Leave any remainder as a fraction
74)
5x3  8x2  7 x  6
3
x
5
Honors Pre-Calculus
75)
10  4 x
Summer Packet 2017
3

 8 x  22x 2  x  5
Page 4 of 5
TOPIC #14 - Complex Number Operations
For problems #76 - #86 , simplify.
76) (3 + 7i) – (6 + 5i)
77) (7 +
 18 ) – (3 + 5 2 i)
2
80) (  50 )
81) (5 + 3i)(4 – 7i)
84) (4 + 7i)(4 – 7i)
85)
3 2 i
4  3i
78)
2
3

1 i 1i
82) (3 – 5i)2
86)
79)
 9   36
83) ( 4   9 )( 2   9 )
8  20 i
2i
Topic # 15 - Exponential and Logarithmic Form
For problems #87 - #89 , write the logarithmic equation into the equivalent exponential form.
87)
88)
89)
1
log
 2
ln 2 = 0.693
log 2 32  5
100
For problems #90 - #92 , write the exponential equation into the equivalent logarithmic form.
90)
91)
92)
34 = 81
10-4 = 0.0001
ex = 5
For problems #93 - #100 , evaluate each expression without using a calculator.
93)
94)
95)
96)
1
log5 1
log4 64
f (x) = log3 x given x =
log4 43
81
97)
98)
99)
100)
log 0.01
ln e
log 1
e ln e
Honors Pre-Calculus
Summer Packet 2017
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