Unit 3 Review for
Common
Assessment
Match the graph of a quadratic function with it’s
equation below:
f(x) = x2
f(x) = -(x+2)2+4
f(x) = (x+2)2-1
Describe the end behavior of the graph of each
given graph.
x  , f (x)  
x   , f ( x )  


x   , f ( x )  
x   , f ( x )  
x   , f ( x )  
x   , f ( x )  


Use the Leading Coefficient Test to determine the end behavior of the graph of
the given polynomial function.
1.) f(x) = -x3 + 4x
Rise Left, fall right
x   , f ( x )  
x   , f ( x )  
EVEN
2.) f(x) = x4 – 5x2 +4
Rise left, rise right

x  , f ( x )  
3.) f(x) = x5 - x
x   , f ( x )  
Fall left, rise right
4.) f(x) = x3 – x2 - 2x
 Fall left, rise right
x  , f ( x )  
x   , f ( x )  
x  , f ( x )  
x   , f ( x )  
5.) f(x) = -2x4 + 2x2
Fallleft, fall right
x   , f ( x )  
x   , f ( x )  
Determine without graphing, the critical points
of each function.
1.) f(x) = (x + 2)2 - 3
f’(x) = -2x + 6
f’(x) = 2x + 4
Min (-2,-3)
3.) f(x) = 3x3 - 9x + 5
f’(x) = 9x2 - 9
2.) f(x) = -x2 + 6x - 8
f’’(x) = 18x
Max (3,1)
4.) = x3 + 6x2 + 5x
Min (-.47, -1.13)
Max (-3.53, 13.12)
Pt. of Inflection (-2,6)
Min ( 1, -1)
Max (-1, 11)
Pt. of Inflection ( 0 , 5)
Min ( -√5, -16)
4
2
5.) f(x) = x - 10x + 9
Max (0, 9)
Min ( √5 , -16)

Find the zeros of each polynomial function.
1.) x2 – 40 = 0
2.) x3 + 4x2 + 4x = 0
x  40
x(x  4 x  4)  0
x 
x ( x  2)( x  2)  0
2
40
x  2 10
3.) x2 + 11x – 102 =0
( x  17 )( x  6)  0
x = -17, 6
If you can’t figure it out then
 use
Quadratic Formula
2
x = 0, -2, -2
4.) x2 + ¾x + ⅛ = 0
x  1 2 x  1 4   0
x = -½, -¼
Find the zeros of the polynomial
function by factoring.
1.) f(x) = x3 + 5x2 – 9x - 45
1.) f(x) = x3 + 4x2 – 25x - 100
x ( x  4 )  25 ( x  4 )  0
2
( x  25 )( x  4 )  0
2
( x  5)( x  5)( x  4 )  0
x = 5, -5, -4

Which of the following is a rational zero of
f(x) = –2x5 + 6x4 + 10x3 – 6x2 – 9x + 4
1, -3, -2, 4, -1 ????
Remember you could use synthetic division or just do p(x) and see if you get
a remainder of ZERO
OR
p( 4 )  2( 4 )  6(4 )  10 ( 4 )  6(4 )  9( 4 )  4
5
4
3
2
So 4 is a factor, the others are not
=0
Use synthetic division to divide x4 + x3 – 11x2 – 5x + 30 by x - 2 . Then
divide by x + 3 Use the result to find all zeros of f(x).
So you are left with:
x2
-5
x2 x
Then all the zeroes are: -3,
C
 5
R
,2
List all possible rational
zeros of
1.)
2.)
List all possible rational roots, use synthetic division to find an actual
root, then use this root to solve the equation.
f(x) = 2x4 + x3 – 31x2 – 26x + 24
Hint 4 and -3/2 are roots
2x2 + 6x – 4
USE QUADRATIC FORMULA!!!
Find the number of possible positive,
negative, and imaginary zeros of:
2,0 positive roots
P
N
2
0
I
P
1 positive root
0
f ( x )  x  4 x  5
f ( x )  x  2 x  x  2 x  2
2
4
0
0
3
2
2
0 negative roots
3,1 negative roots
N
I
1
3
0
1
1
2

3,1 positive roots
f ( x )  6 x  x  4 x  x  2
4
3
1 positive root
P
N
I
P
3,1 positive roots
2
3
1
0
1
1
2

f (  x )  5 x  6 x  24 x  20 x  7 x  2
5
4
3
2,0 positive roots
2
3
3
1
1
N
2
0
2
0
I
0
2
2
4
Use the given root to find the solution
set of the polynomial equation.
p(x) = x4 + x3 – 7x2 – x + 6
GIVEN -3 IS A ROOT
Then we can find the rest by factoring:
x  2x  x  2
3
2
x ( x  2)  1( x  2)
2
( x  1)( x  2)
2
( x  1)( x  1)( x  2)
So the roots are:
-3, -1, 1, and 2
Which equation represents
the graph of the function?
f(x) = 2x2+2x-1
f(x) = -x2-3x+4
f(x) = x2+10x-1
Approximate the real
zeros of each function.
R( x )  3 x  x  1
4
F (x)  x  4 x  6
2
3

0.7, -0.7
H (x)  2 x  4 x  3
3
2.3
-2.5
G(x)  x  3x  1
2
2

-0.4 and -2.6
Use the given root to find the solution set
of the polynomial equations
x  x  8 x  4 x  48
4
3
x  3 x  12 x  54 x  40
2
4
2i
Since 2i is a root, so is -2i
Turn the roots into factors, multiply

them together, then use long
division
( x  2i)( x  2i)  x  4
2
x  x  12
3

x  4 x  x  8 x  4 x  48

4
3
2
3-i
Since 3-i is a root, so is 3+i
Turn the roots into factors, multiply
them together, then use long
division
( x  (3  i))( x  (3  i))  x  6 x  10
2
x  3x  4
2
2
2
2

x  6 x  10 x  3 x  12 x  54 x  40
2
4
3
2
Then factor to find the remaining roots
Then factor to find the remaining roots
x  x  12  ( x  3)( x  4 )
x  3 x  4  ( x  1)( x  4 )
2

So the roots are: 2i, -2i, 3, and -4
2
So the roots are: 3-i, 3+I, 1, and -4
Find the vertical asymptotes, if any, of
the graph of each function.
R( x) 
x
F (x) 
x 4
2
x = -2, x = 2
H (x) 
x

x3
x4
x=4
x 9
2
2
G(x) 
x 1
2
No vertical asymptote

x  4 x  21
2
x = -7
Find the horizontal asymptote,
if any, of the graph of
x
R( x) 
F (x) 
x 4
2
y=0

H (x) 
G(x) 
2
x 1
3x  x
If a monomial is on
bottom then you just
break it up.
2
x  x 1
3
2
2
Otherwise must do long division


3x  3
x  x  1 3x  0 x  x  0 x  0
3
y=1
x4
y=1
4
x
x3
2
4
3
2
y = 3x + 3
Choose the correct graph
for the rational function
x 1
2
R( x) 

x
2x  5x  2
2
F (x) 
x 4
2

H (x) 

x
x 1
2
2( x  2) ( x  5)
2
2
G(x) 
( x  5)( x  2)
2