Section 3.2 Polynomial Functions and Their Graphs
*Definition of a Polynomial Function
Let n be a nonnegative integer and let an , an1 ,, a2 , a1 , a0 be real numbers, with an  0 .
The function defined by
f ( x)  a n x n  a n 1 x n 1    a 2 x 2  a1 x1  a 0
is called a polynomial function of degree n. The number a n , the coefficient of the
variable to the highest power, is called the leading coefficient.
Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.
Example 1) Find the degree and the leading coefficient of each following function.
2
g(x)  3x 2  2x  1 h( x)   x 4  x 6
f (x)  3
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*The Leading Coefficient Test
 or decreases without bound, the graph of the polynomial function
 As x increases
eventually rises or falls. The behavior of a graph to the far left or the far right is called
end behavior. The end behaviors are determined by the degree n and the leading
coefficient an .
1. For n odd (Opposite behavior at each end):
an  0
an  0

y
y


x
x
2. For n even (Same behavior at each end):
an  0
y
an  0
y


x
x
1
Example 2) Use the Leading Coefficient Test to determine the end behavior of the graph
of f (x)  x 4  3x 3  2 .

*Zeros of Polynomial Functions
Zeros: the values of x for which f (x)  0 , i.e. roots or solutions of f (x)  0 .
Each real root of the polynomial appears as an x-intercept of the graph.
Example 3) Find all zeroes of f (x)  x 3  2x 2  4 x  8 .



Example 4) Find all zeroes of f (x)  x 4  4 x 2 .

*Multiplicities of Zeros
In factoring the equation for the polynomial function f, if the same factor x  r occurs k
times, but not k+1 times, we call r a zero with multiplicity k, i.e. (x  r)k appears.
Example 5) Find the zeros and each multiplicity of f (x)  x 2 (x  2)(x  3) 3 .



graph touches the x-axis and turn around at r.
If r is a zero of even multiplicity, then the
If r is a zero of odd multiplicity, then the graph crosses the x-axis at r.
Graphs tend to flatten out at zeros with multiplicity greater than one.
Even Multiplicity
Odd Multiplicity
y
y
x
x
2
Example 6) Find the zeros of f (x)  4 x(x 1)(x  2) 2 and give the multiplicity of each
zero. State whether the graph crosses the x-axis or touch the x-axis and turns around at
each zero.

*Turning Points of Polynomial Functions
If f is a polynomial function of degree n, then the graph of f has at most n 1 turning
points.

*A Strategy for Graphing Polynomial Functions
 end behavior.
Step1 Use Leading Coefficient Test to determine the
Step2 Find x-intercepts by setting f (x)  0 . For a factor (x  r)k of f (x)
a. If k is even, the graph touches the x-axis at r and turns around
b. If k is odd, the graph crosses the x-axis at r.
c. If k>1, the graph flattens out at (r, 0).
 computing f (0) . 

Step 3 Find y-intercept by
Step 4 Use symmetry, if applicable, to help draw the graph:
a. y-axis symmetry: f (x)  f (x)
b. Origin symmetry: f (x)   f (x)

Step 5 Use the fact that the maximum
number of turning points of the graph is n 1 to
check whether it is drawn correctly.

f (x)  x 3  3x 2
Example 7) Graph 


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