Math 130

3.4 – Polynomial Functions: Graphs, Applications, and Models

Graphs of ( )

Ex 1.

Graph ( )

In general, polynomial functions are smooth (i.e. only have rounded curves with no sharp

corners) and continuous (i.e. don’t have breaks and can be drawn without lifting pencil).

Here are graphs that do not come from polynomial functions:

End Behavior

The question “What is the end behavior of a function?” means “What does the function do as

it goes to the far left or far right?” That is, does it go up? Go down? Wiggle up and down?

Get flatter?

For polynomial functions, end behavior depends on the dominating term (term with highest

degree, sometimes called the leading term).

If dominating term …

1. …has odd degree and positive coefficient, then falls to left and rises to right ().

2. …has odd degree and negative coefficient, then rises to left and falls to right ().

3. …has even degree and positive coefficient, then rises to left and rises to right ().

4. …has even degree and negative coefficient, then falls to left and falls to right ().

(Or, just think about what happens to the dominating term when you plug in bigger and bigger

negative #’s, and bigger and bigger positive #’s.)

Ex 2.

Determine the end behavior of the graph of ( )

Ex 3.

Determine the end behavior of the graph of ( )

Zeros and Multiplicity

Zeros of a function ( where ( ) ) correspond with points on the -axis ( -intercepts).

The multiplicity of each zero affects how the graph goes through each -intercept.

If a zero has an odd multiplicity, then the graph crosses the -axis through the -intercept.

If a zero has an even multiplicity, then the graph bounces off the -axis at the -intercept.

Graphing Polynomial Functions

To graph a polynomial function ( ):

1. Find real zeros and plot as -intercepts.

2. Find and plot -intercept.

3. Use test points within intervals made by -intercepts to determine sign of ( ).

4. Use end behavior and multiplicity of zeros to help you draw curve through above points.

Ex 4.

Graph ( )

Intermediate Value Theorem (for Polynomials)

If ( ) defines a polynomial function with only real coefficients, and if for real #’s and , the

values ( ) and ( ) have opposite signs, then there is at least one real zero between and .

Q: What belongs to you but others use it more than you do?