Higher Degree Polynomial Functions and Graphs
Polynomial Function
A polynomial function of degree n in the variable x is a function defined by
P ( x ) where each a i
 a n x n  is real, a n a n

1 x n

1    a
1 x
 a
0

0, and n is a whole number.


 a n is called the leading coefficient
n is the degree of the polynomial a
0 is called the constant term

Polynomial Functions
Polynomial
Function in
General Form y
 ax
 b y
 ax
2  bx
 c y
 ax
3  bx
2  cx
 d y
 ax
4  bx
3  cx
2  dx
 e
Degree
1
2
3
4
Name of
Function
Linear
Quadratic
Cubic
Quartic
The largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Functions
f(x) = 3
ConstantFunction
Maximum
Number of
Zeros: 0

Polynomial Functions f(x) = x + 2
LinearFunction
Degree = 1
Maximum
Number of
Zeros: 1

Polynomial Functions f(x) = x 2 + 3x + 2
QuadraticFunction
Degree = 2
Maximum
Number of
Zeros: 2

Polynomial Functions f(x) = x 3 + 4x 2 + 2
Cubic Function
Degree = 3
Maximum
Number of
Zeros: 3

Polynomial Functions
Quartic Function
Degree = 4
Maximum
Number of
Zeros: 4

Leading Coefficient
The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.
For example, the quartic function f(x) = -2x 4 + x 3 – 5x 2 – 10 has a leading coefficient of -2.

As x increases or decreases without bound, the graph of the polynomial function f ( x )
 a n x n  a n -1 x n -1  a n -2 x n -2  …  a
1 x
 a
0
( a n

0) eventually rises or falls. In particular ,
For n odd: a n
>
0 a n
<
0
If the leading coefficient is positive, the graph falls to the left and rises to the right.
Rises right
Falls left
If the leading coefficient is negative, the graph rises to the left and falls to the right.
Rises left
Falls right

As x increases or decreases without bound, the graph of the polynomial function f ( x )
 a n x n  a n -1 x n -1  a n -2 x n -2  …  a
1 x
 a
0
( a n

0) eventually rises or falls. In particular,
For n even: a n
>
0 a n
<
0
If the leading coefficient is positive, the graph rises to the left and to the right.
Rises right
Rises left
If the leading coefficient is negative, the graph falls to the left and to the right.
Falls left
Falls right

Use the Leading Coefficient Test to determine the end behavior of the graph of f ( x )
 x
3 
3 x
2  x

3.
Rises right y x
Falls left

Determining End Behavior
Match each function with its graph.
A.
f x
 x
4  x
2 
5 x

4 h x

3 x
3  x
2 
2 x

4 g ( x ) k ( x )


 x
6

7 x
7

 x
2 x

3 x

4

4
B.
C.
D.

Quartic Polynomials
Look at the two graphs and discuss the questions given below .
Graph A -5 -4 -3 -2 -1
-2
-4
-6
-8
-10
-12
-14
10
8
6
4
2
1 2 3 4 5
Graph B
-5 -4 -3 -2 -1
-2
-4
-6
-8
-10
14
12
10
8
6
4
2
1 2 3 4 5
1. How can you check to see if both graphs are functions?
2. How many x-intercepts do graphs A & B have?
3. What is the end behavior for each graph?
4. Which graph do you think has a positive leading coeffient? Why?
5. Which graph do you think has a negative leading coefficient? Why?

x -Intercepts (Real Zeros)
Number Of x-Intercepts of a Polynomial Function
A polynomial function of degree n will have a maximum of n xintercepts (real zeros).
Find all zeros of f ( x ) = x 4 + 4 x 3 - 4 x 2 .
 x
4 
4 x
3 
4 x
2  0
We now have a polynomial equation.
x
4 
4 x
3 
4 x
2 
0 Multiply both sides by

1. (optional step) x
2
( x
2 
4 x

4)

0 Factor out x
2
.
x
2
( x

2)
2 
0 x
2  0 or ( x

2)
2 
0
Factor completely.
Set each factor equal to zero.
x

0 x

2 Solve for x.
(0,0) (2,0)

Multiplicity and x -Intercepts
If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph
crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

Example

Find the
x-intercepts and multiplicity of f(x) =2(x+2) 2 (x-3 )

Zeros are at
(-2,0)
(3,0)

Extrema
 Turning points
– where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n

1 is at most n
– 1.
 Local maximum point – highest point or “peak” in an interval
 function values at these points are called local maxima
 Local minimum point – lowest point or “valley” in an interval
 function values at these points are called local minima
 Extrema – plural of extremum , includes all local maxima and local minima

Extrema

Number of Local Extrema





A linear function has degree 1 and no local extrema.
A quadratic function has degree 2 with one extreme point.
A cubic function has degree 3 with at most two local extrema.
A quartic function has degree 4 with at most three local extrema.
How does this relate to the number of turning points?

Comprehensive Graphs


The most important features of the graph of a polynomial function are:
1.
2.
intercepts, extrema,
3.
1.
end behavior.
A comprehensive graph of a polynomial function will exhibit the following features: all x-intercepts (if any),
2.
3.
4.
the y-intercept, all extreme points (if any), enough of the graph to exhibit end behavior.