3.2 Graphs of Polynomial Functions of Higher Degree

Continuous Polynomial Function- no breaks, holes, or gaps;

only smooth rounded turns (no sharp turns like )

xxf

Sketching by hand…

*** Must be in Standard Form***

There will be 2 cases

01

1... axaxaxf nn

Case 1:If n is even (therefore even degree), the graph has a shape similar to

The Right and Left Hand Behavior:

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

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

Case 2:If n is odd (therefore odd degree), the graph has a shape similar to

The Right and Left Hand Behavior:

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.

Describe the right-hand and left-hand behavior.

xxxf

xxxf

xxxf

5

24

3

.3

45.2

4.1

x - a is a factor of

If x = a is a zero, then:x = a is a solution for

Zeros of Polynomial Functions (x values):

For a polynomial of degree n, f has at most n -1 turning points

(where the graph goes from increasing to decreasing and vice versa a.k.a EXTREMA)

and f has at most n real zeros.

0xf xf

(a, 0) is an x intercept of the graph of f.

Find all zeros. 24 22 xxxf

Given a factor of , there is a repeated zero at x = a, of multiplicity k.

• If k is odd, the graph crosses the x axis at x = a • If k is even, the graph touches the x axis at x =

a (bounces)

1, kax k

Steps:

• Determine the right and left hand behavior.• Factor to determine the zeros.• Use the multiplicity factor to determine how

the zeros affects the graph (crosses through or touches x axis).

• Sketch the graph.

Graph Without a Calculator 34 43 xxxf

xxxxf2

962 23

Intermediate Value Theorem

If a<b and , on [a,b], f takes on every value between . This can be used to approximate the real zero.

Example:

bfaf bfaf &

123 xxxf