1

3.2 Polynomial Functions and Their Graphs

A polynomial function of degree n is a function of the form

where n is a nonnegative integer and . The numbers are called the coefficients of the polynomial. The number a0 is the constant coefficient or constant term. The number an, the coefficient of the highest power, is the leading coefficient, and the term anxn is the leading term.

2

The table lists some more examples of polynomials.

If a polynomial consists of just a single term, then it is called a monomial. For example, P (x) = x3 and Q (x) = –6x5 are monomials.

Here's another monomial

3

Give the degree, leading term, leading coefficient, coefficients, and constant term of the following polynomial:

4

The simplest functions are the monomials P(x)=xn.

Again, we can go to page 202 of our text to see these

power functions.

Even Powers Odd Powers

5

We can use transformations on these graphs to sketch the graphs of the following functions:

6

Sketch the graphs of the following functions:

7

The Graph of a Polynomial function is continuous. This means that the graph has no breaks or holes. It also is a smooth curve; that is, it has no corners or sharp points (cusps)

Not polynomial Not polynomial Polynomial Polynomial

8

Do you remember end behavior of polynomial graphs?

9

10

The end behavior of a function is a description of what happens as x becomes large in the positive or negative direction.

For example the monomial has the following end behavior

andbook notation:

This line describes the right side of the graph: "y goes up as x goes to the right."

This line describes the left side of the graph: "y goes up as x goes to the left."

Or you can write it this way:

11

State the end behavior of

12

Determine the end behavior of the polynomial

13

Determine the end behavior of the polynomial

14

15

Sketch the graph of the polynomial function

(Make sure to find the end behavior and zeros.)

16

Let (a) State the end behavior(b) Find the zeros of P(c) Sketch the graph of P

17

Odd Multiplic

ity

Even 

Multiplicity

18

Sketch the graph

19

Sketch the graph

20

Sketch the graph (pay attention to multiplicity)

0

21

Sketch the graph

Since the power on the second term is half of the leading power, then this function is in "quadratic form" and we solve it very similarly to a regular quadratic as shown below.

This one doesn't give us any real solutions

22

Find all x & yintercepts and local extrema:

23

From Your Homework:

24

From Your Homework: State end behavior and match each function to its graph.

25

              Assignment:Section 3.2: 

problems 153 odd (for 4549 only state the end behavior of each function)

Attachments

powers of x.gcx

SMART Notebook

Page 1: Aug 3-8:28 AMPage 2: Feb 2-7:40 AMPage 3: Feb 2-7:41 AMPage 4: Aug 3-8:44 AMPage 5: Aug 3-8:46 AMPage 6: Aug 3-8:53 AMPage 7: Aug 3-8:08 AMPage 8: Feb 2-9:09 AMPage 9: Feb 2-7:50 AMPage 10: Aug 3-8:55 AMPage 11: Aug 3-9:05 AMPage 12: Aug 3-9:21 AMPage 13: Aug 3-9:23 AMPage 14: Aug 3-9:28 AMPage 15: Aug 3-9:26 AMPage 16: Aug 3-9:39 AMPage 17: Feb 2-7:52 AMPage 18: Aug 3-9:42 AMPage 19: Aug 3-9:43 AMPage 20: Aug 3-9:46 AMPage 21: Feb 2-9:20 AMPage 22: Feb 2-9:43 AMPage 23: Feb 2-9:43 AMPage 24: Feb 2-9:24 AMPage 25: Feb 2-11:06 AMAttachments Page 1