Polynomial Functions and Graphs

Higher Degree Polynomial Functions and Graphs

an is called the leading coefficient n is the degree of the polynomial a0 is called the constant term

Polynomial Function

A polynomial function of degree n in the variable x is a function defined by

where each ai is real, an 0, and n is a whole number.01

11)( axaxaxaxP n

nn

n

Polynomial Functions

The largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Function in

General FormDegree

Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quarticedxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy

Polynomial Functions

f(x) = 3

ConstantFunction

Degree = 0

Maximum Number of

Zeros: 0

f(x) = x + 2LinearFunction

Degree = 1

Maximum Number of

Zeros: 1

Polynomial Functions

f(x) = x2 + 3x + 2QuadraticFunction

Degree = 2Maximum Number of

Zeros: 2

Polynomial Functions

f(x) = x3 + 4x2 + 2

Cubic Function

Degree = 3

Maximum Number of

Zeros: 3

Polynomial Functions

Quartic Function

Degree = 4

Maximum Number of

Zeros: 4

Polynomial Functions

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) = -2x4 + x3 – 5x2 – 10 has a leading

coefficient of -2.

The Leading Coefficient Test

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

f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)

eventually rises or falls. In particular,

For n odd: an 0 an 0

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

f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)

eventually rises or falls. In particular,

For n odd: an 0 an 0

If the leading coefficient is positive, the graph falls to the left and rises to the right.

If the leading coefficient is negative, the graph rises to the left and falls to the right.

Rises right

Falls left

Falls right

Rises left

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

f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)

eventually rises or falls. In particular,

For n even: an 0 an 0

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

f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)

eventually rises or falls. In particular,

For n even: an 0 an 0

If the leading coefficient is positive, the graph rises to the left and to the right.

If the leading coefficient is negative, the graph falls to the left and to the right.

Rises right

Rises left

Falls left

Falls right

The Leading Coefficient Test

Example

Use the Leading Coefficient Test to determine the end behavior of the graph of f (x) x3 3x2 x 3.

Falls left

yRises right

x

Determining End Behavior

Match each function with its graph.

4 2

3 2

( ) 5 4

( ) 3 2 4

f x x x x

h x x x x

47)(

43)(7

26

xxxkxxxxg

A. B.

C. D.

Quartic Polynomials

Look at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

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?

2. How many x-intercepts do graphs A & B have?

Graph BGraph A-5 -4 -3 -2 -1 1 2 3 4 5

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

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 + 4x3 - 4x2.x4 4x3 4x2 We now have a polynomial equation.

x4 4x3 4x2 0 Multiply both sides by 1. (optional step)

x2(x2 4x 4) 0 Factor out x2.

x2(x 2)2 0 Factor completely.

x2 or (x 2)2 0 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. 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.