For those going into Pre-Calculus
Franklin High School Summer Math Packet
Summer Brain Drain: Did you know that research has found that students lose on average 2.6 months of their
math skills over the summer months? This is almost 1/3 of a school year! (today.com)
To help combat summer brain drain, we are requiring you to complete a summer math packet to help you
maintain your math skills. This packet will count as your first test grade for the new school year. You will be
responsible for handing in the completed packet ON THE FIRST DAY OF SCHOOL. If there is not enough
space provided, work problems on binder paper. Make sure that all pages and problems are numbered correctly.
Show all of your work. Write neatly. BOX all final answers.
Use the following websites if you need help with the content: (and there are plenty more websites)
http://www.khanacademy.org/
http://www.ixl.com/math
http://hotmath.com/hotmath_help/topics/index_hotmath_review_full.html
These websites will help you practice your math skills: (and there are plenty more websites)
www.hoodamath.com
www.coolmath.com
www.mathplayground.com
www.multiplication.com/games
PLEASE BUY A CALCULATOR THIS SUMMER!
6th Grade – Geometry: any scientific calculator
Algebra 3-4 and higher: we recommend a TI-83 or TI-84
This packet should be completed independently and your parent or guardian needs to sign off on this page
UPON YOUR COMPLETION of this packet.
I understand that if I do not complete this packet, I will receive a zero for my first test grade and will jeopardize
my ability to pass the class.
Student signature:
Parent/Guardian signature:
F.H.S.
Mathematics
PrePre-Calculus Summer Review Guide
1.) Geometry Topics
Equations of a line:
1. Slope intercept: y = mx + b
where
slope =
y 2 − y1
x 2 − x1
2. Point slope: y – y1 = m(x – x1)
3. Standard: Ax + By + C = 0
Directions: Solve each problem in the space provided, circling your final answer. Put all answers in
standard form.
1. Write the equation of the line parallel to 2x – 6y = -1 and containing the x-intercept of
4x – 3y = 12.
2. Write the equation of the line in slope intercept form through the point with coordinates (-4,6) and
perpendicular to 3x – 2y = 8.
3. Find the value of “a” if a line containing the point (a, -2a) has a y-intercept of 6 and slope of -2/3.
4. Write the equation of the perpendicular bisector of the segment joining the points with coordinates of
(-3,4) and (5,-2).
2.) Rules of Exponents
Properties:
a ⋅a = a
m
a
−n
n
(a )
m n
m+n
=a
p
r
a = r ap
mn
m
am
= a m −n
an
am
a
  = m
b
b
1
= n
a
Directions: Simplify each in the space provided, showing all steps. Answers should have positive
exponents. Circle your answer.
2
0
1. (2x y) (3xy)
2. a b a
(a b c )
(ab c )
−3
-2
3
4. (2x) (2y) (4x)
2
 5u 2 v   − 3uv 

7. 
2  
2 
2
uv

  2u v 
4 −5 4 6
3.
42
-2 3 3
5.
2
−2
− 2 3 −1
2 4 8 316
6.
32 −1
3
8. (3-1 + 2-1)2
9.
−2
a −1 − 3a −2
2 −2
3.) Factoring
Strategies to use:
1. Greatest Common Factor (GCF)
2. Difference of Squares
3. Trinomials
4. Sum and Difference of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
5. Grouping
Directions: Factor each of the following completely, circling your final answer.
Trinomials
a.
2x2 + 5x + 3
b.
3x2 + 7x + 2
c.
5x2 - 7x + 2
d.
6x2 - 11x + 3
e.
6x2 – 13x – 5
f.
4x2 – 11x – 3
g.
7x2 + 9x + 2
h.
7x2 – 10x + 3
i.
2x2 – 9x – 5
j.
2x2 + xy - 6y2
k.
3y2 – 17xy – 6x2
l.
8x2 – 27x – 20
b.
225x4 – 64y8
c.
16a4 – 81y8
Differences of perfect squares.
a.
9x2 – 16y2
Sums and difference of perfect cubes.
a.
27x3 + 1
c.
y6 + 216
b.
8a3 – y3
d.
m6 - 2
Combinations of perfect squares and cubes.
a.
x6 – 1
b.
x6 + y12
c.
x12 – y 24
2x9 + 10x6 + 12x3
c.
x8 – 5x4 – 36
Polynomials in quadratic form
a.
16x4 – 40x + 25
b.
Factor by grouping
a.
2ax + 6xc + ba + 3bc
b.
3my + 7x + 7m + 3xy
c.
a2 - 2ab + a - 2b
d.
4ax - 14bx + 35by - 10ay
e.
x3 + 2x2 - x – 2
f.
x2 + 6x + 9 - a2
g.
n2 + 2nx - 1 + x2
h.
b2 - y2 - 2yp - p2
i.
a2 + 2ab + b2 – 9
j.
x3 + y3 - x2y - xy2
4.) Function Notation
Directions: Find the value of each in the space provided, showing all steps. Circle your answer.
Given: f(x) = 3x – 7
g(x) = x2 + 3
1. Find f(-1)
2. f (x + 3)
3. f(f(x))
4. g(x + 2) – g(x)
5. f(g(2))
6. g(f(2))
5.) Rational Expressions
Simplify the following expressions :
1.
x
xy
3.
5.
2
−
2x
x
2
x 2 − 5x + 6
x−2
2x 2 + x − 6
x 2 + 4x − 5
•
x 3 − 3x 2 + 2 x
4x 2 − 6x
2.
x
2
+
x − 3 3x + 4
4.
1− x
x −1
6.) Simplify Complex Fractions
When simplifying complex fractions, multiply both the numerator and denominator by the reciprocal
of the denominator. Remember to also look for common factors to simplify.
1.
x2
x −1
2x
x −1
x 2 + 2x +1
3.
x2 − 4
x +1
x2 − x −6
2.
3
−4
x +1
2x
x +1
7.) Radicals
Simplify radicals whenever possible:
ex1)
x5 y 7 z 6
ex2)
x2 x2 x⋅ y2 y2 y2 y ⋅ z2 z2 z 2
x⋅x⋅ y⋅ y⋅ y⋅z⋅z⋅z
x2 y3z3
ex3)
x• y
3x ⋅ 4 x
12 x
3
10 x
12 x
4
10
− 32
3
− 2 ⋅ −2 ⋅ −2 ⋅ 2 ⋅ 2
− 23 4
xy
3x 2 x • 4 x 2 5 x
2
3
ex4)
2x ⋅ 5x
5 27 + 4 2 + 7 3
15 3 + 4 2 + 7 3
22 3 + 4 2
2
Rationalize fractions with radicals in the denominator:
ex5)
24
15
ex6)
8
5
2
4
4
1
2⋅2⋅2⋅2⋅2
2− 3
2⋅ 2
4
•
1
5
4
5
2 10
5
, now rationalize
2
•
4
4
22 −
2
1
4
2+ 3
•
2+ 3
2+ 3
2
5
5
1
2− 3
2
2 2
2 2
ex7)
32
4
2⋅2⋅2
4
2⋅2⋅2
( 3)
2
2+ 3
4−3
2+ 3
8
2⋅2⋅2⋅2
4
8
2
Simplify the following:
1.
3
2.
24
Rationalize the following:
5
5.
2 3
3
− 40 x 6 y 7
6.
3.
2
3
75 x 3 • 5 x 3
2 48 − 3 27
4.
7.
5
2
3+ 7
True or False – Explain why.
8.
x=
x for all x
9.
a2 + b2 = a + b
10.
4
16 = 4 − 16
8.) Basic Polynomial Graphs -- Translations/Transformations of Parent Functions
Be able to quickly sketch these five basic “parent” graphs from memory:
y = x2
y= x
y= x
y = x3
y=
1
x
y = log b ( x )
To translate or move a parent function, use these basic examples:
the equation y = ( x − 5)2 − 3 moves the graph 5 units to the right and 3 units down
the equation y = (x + 2 )2 + 4 moves the graph 2 units to the left and 4 units up
To transform or stretch/shrink a parent function, use these basic examples:
1
1
as tall as the parent graph.
the equation y = x 2 shrinks the graph vertically so the height is
3
3
the equation y = 4 x 2 stretches the graph vertically so the height is 4 times as tall as the parent graph.
3
1 
the equation y =  x  stretches the graph horizontally so it is 2 times as wide as the parent graph.
2 
1
the equation y = (3x )3 shrinks the graph horizontally it is as wide as the parent graph.
3
Use the equations below to ACCURATELY sketch the graphs the functions without the aid of your calculator. Please label at least 3
points, including vertices, and also label the asymptotes if necessary.
1
1.
2.
3.
y = x −3 +2
y = x+2
y=
−2
x+2
4.
y=
1 2
x
2
5.
y = 3( x − 1) 3
6.
y = log 4 ( x + 2 )