Polynomial Functions and their
Graphs

1. Identify polynomial functions.
2. Recognize characteristics of polynomial functions.
3. Determine end behavior.
4. Identify zeros of a polynomial function (and their multiplicities).
5. Use the Intermediate Value Theorem.
6. Graph polynomial functions.

Let n be a nonnegative integer and let a n a n-2
, … , a
2
, a
1
, a
0 be real numbers with a equal to zero. The function defined by n
, a n-1 not
, f ( x )
 a n x n  a n

1 x n

1  a n

2 x n

2 
...
 a
2 x
2  a
1 x
 a
0 is called a polynomial function of degree n.
Further, a n is called the leading coefficient.
Now, before you say, “I’ve never seen this before…”

• y = x 2 is a polynomial function. It has a degree of 2 and the leading coefficient is 1. Second degree polynomial functions are quadratic.
• So also is y = 2x – 5. It has a degree of
(understood) 1 and the leading coefficient is 2.
First degree polynomial functions are linear.
• Even y = 5 is a polynomial function. The degree is 0. We call this type of polynomial function constant.

• Third-degree polynomial functions are called cubics.
• Fourth-degree polynomial functions are called quartics.
• Fifth-degree polynomial functions are called quintics.
• Sixth-degree and higher do not have special names.

• The exponents of a polynomial function will all be nonnegative integers (no fractions or decimals).
• The graphs of polynomial functions are smooth and continuous:
 “smooth” means rounded curves with no sharp corners.
 “continuous” means the graphs have no breaks.

f ( x )

4 x
4 
3 x
3 h ( x )

5 x
4 
2 x
2 
1 x g ( x )
 x
1 3 
4 x

7

• The behavior of the graph to the far right (as x approaches ∞) or the far left (as x approaches
-∞).
• End behavior is determined by two factors:
1. The degree of the polynomial function (even or odd)
2. The sign of the leading coefficient (positive or negative)

The degree of the polynomial function is even.
The degree of the polynomial function is odd.
The leading coefficient is positive.
The leading coefficient is negative.

f ( x )

6 x
7 
3 x
6 
5 x
5 
5 f ( x )
 
3 x
4  x
3 
5 x
2 
6 x

9 f ( x )

2 x
2 
6 x

5 f ( x )
 
4 x
5 
4 x
4  x
2 
1

• In order to determine the behavior of the graph between the ends, we need to find two things:
1. The zeros of the function
2. The multiplicity of each zero

• To find the zeros of a polynomial function, set the function equal to zero and solve for x.
• WARNING!!!! Factoring may be required.
• Recall that, when solving by factoring, several techniques may be employed:
 Greatest common factor
 Factor by grouping
 Trinomial factoring (AC method or trial-and-error)
• Then, set each factor equal to zero.

• The multiplicity of a zero is the number of times that value is a zero of the polynomial.
• We use exponents to determine multiplicity.
• The multiplicity determines whether the graph crosses the x-axis at that value (odd multiplicity) or touches and turns around
(even multiplicity) at that value.

• For each function below, find the zeroes, their multiplicity, and state the behavior of the graph at the respective zeroes.
f ( x )
 
4 ( x

7 )( x

9 )
2 f ( x )
 x
3 
4 x
2 
4 x f ( x )
 x
3 
6 x
2 
4 x

24

1. Use the degree and the leading coefficient to determine end behavior.
2. Find all zeroes (x-intercepts) and their multiplicities. Determine the behavior of the graph at each zero (does the graph cross, or does it touch and turn around?
3. Find the y-intercept by setting x = 0.