LESSON 14 (2.2) GRAPHS OF POLYNOMIAL FUNCTIONS
You should learn to:
1. Use the leading coefficient to determine the end behavior for graphs of polynomial functions.
2. Find and use zeros of polynomial functions as sketching aids.
Terms to know: monomial, end behavior (of a graph), leading coefficient (Leading Coefficient Test),
term versus factor, zeros of a function, extrema, linear factor, multiplicity, continuous
A polynomial function is a function with many terms. In this section, we will be looking at polynomial functions
of a higher degree.
The graph of a polynomial function is a smooth and continuous curve.
Example 1: Graph and compare the following.
y=x
y  x3
y  x2
y  x4
y  x5
y  x6
End Behavior and the Leading Coefficient Test:
Note: You must make sure that the polynomial is in term form (NOT
factored form) to use the Leading Coefficient Test.
Even
Odd
+
 
 
–
 
 
Example 2: Find the end behavior for the following functions:
a. y  2 x3  4 x 2  1
 
d.
y  ( x  2)( x  1)( x  5)
( x3 )
 
b. y  5x 4  7 x
 
c. f  x    x 4  x3  2 x 2
e. y  ( x  3)3 ( x  1)2
( x5 )  
f. y   x  3 x  2   x  1
 
2
( x6 )
 
3
The graph of a polynomial function (of x ) has exactly one y-intercept. To find a y-intercept, set x = 0.
The graph of a polynomial function of degree n has at most n x-intercepts. A zero of a function f is a
number x which makes f  x   0 . Real zeros of polynomial functions are the x-values of the x-intercept
points.
Example 3: Find the zeros for each of the polynomial functions.
a.
f ( x)  x 3  x 2  2 x
c. y   x  3 x  2 
b. g ( x)  x 2  x  3
2
 x  1
3
f ( x)  x 3  x 2  2 x
g ( x)  x 2  x  3
y   x  3 x  2   x  1
0  x3  x 2  2
0  x2  x  3
0   x  3 x  2   x  1
0  x( x 2  x  2)
0  x( x  2)( x  1)
x
x  0, x  2, x  1
2
2
(1)  (1) 2  4(1)(3)
2(1)
3
3
x  3, x  2, x  1
1  1  12
2
1  13
x
2
x  2.303, x  1.303
x
x-intercepts with Multiplicity:
Even: Bounce
Odd: Cross
The graph has at most n 1 relative extrema (relative minimum or maximum values - that is, turning
points.)
Steps to Graph Polynomial functions
1. Remember they are continuous (there are no breaks in their graphs) and have only smooth turns (there are
no sharp turns).
2. Find end behavior determined by the Leading Coefficient Test.
4. Find the y-intercept by setting x = 0
*5. Find x-intercepts by setting f  x   0 and factoring or using the quadratic formula.

crosses at  a, 0  if k is odd

bounces at  a, 0  if k is even
*For a factor  x  a  : the graph 
k
Example 4: Graph the following functions.
a. g ( x)  3x3  12 x 2
End behavior  
y-intercept (0, 0)
g ( x)  3 x 3  12 x 2
0  3 x3  12 x 2
0  3 x 2 ( x  4)
x  0, x  4
(0,0) bounces, (4,0) crosses
x-intercepts
b.
f ( x)  x 4  2 x 2  1
End behavior  
y-intercept (0,1)
f ( x)  x 4  2 x 2  1
0  x4  2 x2  1
0  ( x 2  1)( x 2  1)
0  ( x  1)( x  1)( x  1)( x  1)
0  ( x  1) 2 ( x  1) 2
x  1, x  1
x-intercepts
(1,0) bounces, (1,0) crosses
c. f ( x)  ( x  1) 2 ( x  2) 4 ( x  3)
Degree:
7
( x 7 )
End behavior:  
x -intercepts: (1,0) bounces, (2,0) bounces, (3,0) crosses
If you are given the zeros (or x-intercepts) of a polynomial function, you can build a possible polynomial function
by building a factor from each zero.
Example 5: Write a polynomial function having zeros of x  3 , and x  3 (multiplicity 2).
f  x   ( x  3)( x  3)( x  3)
f ( x)  ( x 2  9)( x  3)
f ( x)  x3  3x 2  9 x  27
b. Simplify your polynomial equation.
ASSIGNMENT 14
Pages 112-115 (Vocabulary Check 1, 2, 5; 1-9, 13, 15-18, 20, 22 (show the end
behavior using directed arrows), 23, 26, 30, 33 (quadratic formula),
38, 48, 51 (simplify-expand), 70 (graph without using a calculator - you will need
to factor first. See Example 4), 85, 99-104, 109, 111, 112, 114) Page 99 18,32
Extra Question 1: Graph y   x( x  1)( x  2)2 without using a calculator.
2: Graph y  x 2 (1  x)(3  x) without using a calculator.