6.8 Analyzing Graphs of Polynomial Functions

Zeros, Factors, Solutions, and Intercepts.

Let1

1 1 0( ) ...n nn nf x a x a x a x a

be a polynomial function.

The following statements are equivalent:

Zero: K is a zero of the polynomial function f.

Factor: (X – k) is a factor of the polynomial f(x).

Solution: K is a solution of the polynomial equation f(x) = 0.

If k is a real number, then the following is also equivalent:

X-intercept: K is an x-intercept of the graph of the polynomial function.

Graph the function: 21

( ) ( 2)( 1)4

f x x x

Solution:

Since (x+2) and (x-1) are factors…

-2 and 1 are x-intercepts…

4

3

2

1

-1

-2

-3

-4

-3 -2 -1 1 2 3

Plotting a few more points will determine the complete graph.

Graph the function: 21

( ) ( 2)( 1)4

f x x x

Solution:

Since (x+2) and (x-1) are factors…

-2 and 1 are x-intercepts…

4

3

2

1

-1

-2

-3

-4

-3 -2 -1 1 2 3

We will call these points turning points.

The turning points of the graph of a polynomial function are another important characteristic of the graph.

Local maximum

Local mimimum

Turning Points of Polynomial Functions

The graph of every polynomial function of degree n has at most n – 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then it’s graph has exactly n – 1 turning points.

You can use a graphing calculator to find the maximums and minimums of quadratic functions. If you take calculus, you will learn analytic techniques for finding maximums and minimums.

Use the graphing calculator to:

*Identify the x-intercepts

*Identify where the local maximum and minimum occur.

3 2( ) 3 2f x x x