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) = an x + an−1 x

n

n −1

+ + a1 x + a0

where each ai is real, an ≠ 0, and n is a whole number.

an is called the leading coefficient

n is the degree of the polynomial

a0 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

Name of

Function

1

2

3

4

Linear

Quadratic

Cubic

Quartic

The largest exponent within the

polynomial determines the degree of the

polynomial.

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) = -2×4 + x3 – 5×2 – 10 has a leading

coefficient of -2.

The Leading Coefficient Test

As x increases or decreases without bound, the graph of the

polynomial function

n

n-1

n-2

f (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ≠ 0)

n

n-1

n-2

1

0

n

eventually rises or falls. In particular,

For n odd:

If the

leading

coefficient is

positive, the

graph falls

to the left

and rises to

the right.

an > 0

n

Rises right

Falls left

an < 0

n

If the

leading

coefficient is

negative, the

graph rises

to the left

and falls to

the right.

Rises left

Falls right

The Leading Coefficient Test

As x increases or decreases without bound, the graph of the polynomial

function

n

n-1

n-2

f (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ≠ 0)

n

n-1

n-2

1

0

n

eventually rises or falls. In particular,

For n even:

an > 0

an < 0

n

n

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

Example

Use the Leading Coefficient Test to determine the

end behavior of the graph of f (x) = x3 + 3×2 − x − 3.

y

Rises right

x

Falls left

Determining End Behavior

Match each function with its graph.

f ( x) = x − x + 5 x − 4

h( x ) = 3 x 3 − x 2 + 2 x − 4

4

A.

C.

2

g ( x ) = −x 6 + x 2 − 3 x − 4

k ( x ) = −7 x 7 + x − 4

B.

D.

Quartic Polynomials

Look at the two graphs and discuss the questions given

below.

10

14

8

12

6

10

4

8

2

Graph A

-5

-4

-3

-2

-1

-2

-4

-6

6

1

2

3

4

5

Graph B

4

2

-5

-4

-3

-8

-10

-12

-14

-2

-1

-2

1

2

3

4

5

-4

-6

-8

-10

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 x- intercepts (real zeros).

Find all zeros of f (x) = -x4 + 4×3 – 4×2.

−x4 + 4×3 − 4×2 = 0

x4 − 4×3 + 4×2 = 0

x2(x2 − 4x + 4) = 0

x2(x − 2)2 = 0

x2 = 0

x=0

or

(x − 2)2 = 0

x=2

We now have a polynomial equation.

Multiply both sides by −1. (optional step)

Factor out x2.

Factor completely.

Set each factor equal to zero.

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.

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

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:

intercepts,

2.

extrema,

3.

end behavior.

A comprehensive graph of a polynomial function

will exhibit the following features:

1.

all x-intercepts (if any),

2.

the y-intercept,

3.

all extreme points (if any),

4.

enough of the graph to exhibit end

behavior.

1.

Editor’s Notes

Teachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily.