2.2 Polynomial Functions 2015/16 Digital Lesson. HWQ 8/17 Copyright © by Houghton Mifflin Company,...

2.2Polynomial Functions

2015/16

Digital Lesson

HWQ 8/17

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

2

Complete the square on the quadratic

and state the vertex.

2 12 17y x x

Polynomial Functions are Continuous and Smooth

y

x–2

2y

x–2

2

y

x–2

2

Functions that are not continuous are

not polynomial functions

(Piecewise)

Functions that have sharp turns are not

polynomial functions (Absolute Value)

Polynomial functions have graphs that are

continuous and smooth

• The polynomial functions that have the simplest graphs are monomials of the form

• If n is even-the graph is similar to

• If n is odd-the graph is similar to

• For n-odd, the greater the value of n, the flatter the graph near(0,0)

y

x–2

2

0,)( nxxf n

2( )f x x

3( )f x x

Transformations of Monomial Functions

Example 1:

5)( xxf

The degree is odd, the negative coefficient reflects the graph on the x-axis, this graph is similar to

3)( xxf

Transformations of Monomial FunctionsExample 2:

The degree is even, and has as upward shift of one unit of the graph of

4)( xxf

4( ) ( 1) 1h x x


A polynomial function is a function of the form1

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

where n is a nonnegative integer and each ai (i = 0, , n)

is a real number. The polynomial function has a leading coefficient an and degree n.

Examples: Find the leading coefficient and degree of each polynomial function.

Polynomial Function Leading Coefficient Degree5 3( ) 2 3 5 1f x x x x

3 2( ) 6 7f x x x x 5

The Leading Coefficient Test (End Behavior Test)

( ) 1 4f x

2

14

3

0

• The graph of a polynomial eventually rises or falls. • This can be determined by the function’s degree (odd or even)

and by its leading coefficient (positive or negative)

y

x

–2

2

When the degree is odd:

If the leading coefficient is

positive

The graph falls to the left and rises

to the right


negative

The graph rises to the

left and falls to the right

y

x

–2

2

When the degree is even:


positive

The graph rises to the left and rises

to the right


negative

The graph falls to the

left and falls to the right


A real number a is a zero of a function y = f (x)if and only if f (a) = 0.

A polynomial function of degree n has at most n real zeros.

Real Zeros of Polynomial Functions

If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.

1. x = a is a zero of f.

2. x = a is a solution of the polynomial equation f (x) = 0.

3. (x – a) is a factor of the polynomial f (x).

4. (a, 0) is an x-intercept of the graph of y = f (x).

Zeros of Polynomial Functions

The graph of f has at most n-1 relative extrema (relative minima or maxima.)


0,)( nxxf n

A polynomial function of degree n has at most n real zeros. It has exactly n total zeros (both real and imaginary.)

Use the leading coefficient test (end behavior test)to describe left and right hand behavior and sketch the graph

Example 1

xxxf 4)( 3

y

x

–2

2


Example 2

45)( 24 xxxf

y

x

–2

2


Example 3

xxxf 5)(

y

x

–2

2


y

x–2

2

Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.

Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).

The real zeros are x = –1, x = 0, and x = 2.

Notice that there is a zero at x = 0 that has a multiplicity of 2. A zero with an even multiplicity will bounce off the x-axis.

f (x) = x4 – x3 – 2x2

(–1, 0) (0, 0)

(2, 0)

Finding Zeros of a Polynomial Function

• Student ExampleFind all real zeros of


xxxxf 2)( 23

The zeros are 0, 2, 1x x x

The x-intercepts are 0,0 , 2,0 , 1,0

Example continued:Sketching the graph of a Polynomial Function with known zeros:

Sketch a graph by hand.


y

x

–2

2

xxxxf 2)( 23

The x-intercepts are 0,0 , 2,0 , 1,0

Finding a Polynomial Function with Given Zeros

Write an equation for a polynomial function with zeros at x = -2, 1, and 3. Sketch a graph by hand.


y

x

–2

2

Finding a Polynomial Function with Given Zeros

Student Example: Find a polynomial function

with the given zeros: x = -1, 2, 2

Sketch a graph by hand.


y

x

–2

2

Sketching the Graph of a Polynomial Function

Sketch the graph of

1. What is the end behavior?

2. Find the zeros of the polynomial function.

3. Plot a few additional points.

4. Draw the graph.


3 2( ) 3 9f x x x

y

x

–2

2

Sketching the Graph of a Polynomial Function

Sketch the graph of

1.What is the end behavior?

2.Find the zeros.

3.Plot a few additional points.

4.Draw the graph.


343)(4

xxxf

y

x

–2

2

Homework

• Section 2.2 pg. 108

1-7 odd, 17-43 odd, 49-55 odd, 61


2.2 Polynomial Functions 2015/16 Digital Lesson. HWQ 8/17 Copyright © by Houghton Mifflin Company,...

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