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OBJECTIVES:

1. Recognize and evaluate polynomial functions. 2. Use a polynomial function to model data. 3. Add and subtract polynomial functions. 4. Graph basic polynomial functions

1. Recognize and evaluate polynomial functions. 2. Use a polynomial function to model data. 3. Add and subtract polynomial functions. 4. Graph basic polynomial functions

1.
Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 6.3 - 1

Sec 6.3 - 1

2.
Chapter 6

Exponents, Polynomials, and

Polynomial Functions

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Sec 6.3 - 2

Exponents, Polynomials, and

Polynomial Functions

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 6.3 - 2

3.
6.3

Polynomial Functions

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Sec 6.3 - 3

Polynomial Functions

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Sec 6.3 - 3

4.
6.3 Polynomial Functions

Objectives

1. Recognize and evaluate polynomial functions.

2. Use a polynomial function to model data.

3. Add and subtract polynomial functions.

4. Graph basic polynomial functions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 4

Objectives

1. Recognize and evaluate polynomial functions.

2. Use a polynomial function to model data.

3. Add and subtract polynomial functions.

4. Graph basic polynomial functions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 4

5.
6.3 Polynomial Functions

Definition of a Polynomial Function

Polynomial Function

A polynomial function of degree n is defined by

f (x) = an xn + an – 1 xn – 1 + · · · + a1 x + a0 ,

for real numbers an,an – 1, . . . , a1, and a0 , where an ≠ 0 and n is a whole

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 5

Definition of a Polynomial Function

Polynomial Function

A polynomial function of degree n is defined by

f (x) = an xn + an – 1 xn – 1 + · · · + a1 x + a0 ,

for real numbers an,an – 1, . . . , a1, and a0 , where an ≠ 0 and n is a whole

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 5

6.
6.3 Polynomial Functions

EXAMPLE 1 Evaluating Polynomial Functions

Let f(x) = 4x3 – 5x2 + 7. Find each value.

(a) f(2)

f(x) = 4x3 – 5x2 + 7

f(2) = 4 • 23 – 5 • 22 + 7

= 4•8 – 5•4 + 7

= 32 – 20 + 7

= 19

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 6

EXAMPLE 1 Evaluating Polynomial Functions

Let f(x) = 4x3 – 5x2 + 7. Find each value.

(a) f(2)

f(x) = 4x3 – 5x2 + 7

f(2) = 4 • 23 – 5 • 22 + 7

= 4•8 – 5•4 + 7

= 32 – 20 + 7

= 19

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 6

7.
6.3 Polynomial Functions

EXAMPLE 1 Evaluating Polynomial Functions

Let f(x) = 4x3 – 5x2 + 7. Find each value.

(b) f(–3)

f(x) = 4x3 – 5x2 + 7

f(–3) = 4 • (–3)3 – 5 • (–3)2 + 7

= 4 • (–27) – 5•9 + 7

= –108 – 45 + 7

= –146

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 7

EXAMPLE 1 Evaluating Polynomial Functions

Let f(x) = 4x3 – 5x2 + 7. Find each value.

(b) f(–3)

f(x) = 4x3 – 5x2 + 7

f(–3) = 4 • (–3)3 – 5 • (–3)2 + 7

= 4 • (–27) – 5•9 + 7

= –108 – 45 + 7

= –146

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 7

8.
6.3 Polynomial Functions

Functions

While f is the most common letter used to represent functions, recall that

other letters such as g and h are also used. The capital letter P is often used

for polynomial functions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 8

Functions

While f is the most common letter used to represent functions, recall that

other letters such as g and h are also used. The capital letter P is often used

for polynomial functions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 8

9.
6.3 Polynomial Functions

EXAMPLE 2 Using a Polynomial Model to Approximate

Data

The number of U.S. households estimated to see and pay at least one bill

on-line each month during the years 2000 through 2006 can be modeled by

the polynomial function defined by

P(x) = 0.808x2 + 2.625x + 0.502,

where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and

so on, and P(x) is in millions. Use this function to approximate the number

of households expected to pay at least one bill on-line each month in 2006.

Since x = 6 corresponds to 2006, we must find P(6).

P(x) = 0.808x2 + 2.625x + 0.502

P(6) = 0.808(6)2 + 2.625(6) + 0.502 Let x = 6.

= 45.34 Evaluate.

Thus, in 2006 about 45.34 million households are expected to pay at least

one bill on-line each month.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 9

EXAMPLE 2 Using a Polynomial Model to Approximate

Data

The number of U.S. households estimated to see and pay at least one bill

on-line each month during the years 2000 through 2006 can be modeled by

the polynomial function defined by

P(x) = 0.808x2 + 2.625x + 0.502,

where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and

so on, and P(x) is in millions. Use this function to approximate the number

of households expected to pay at least one bill on-line each month in 2006.

Since x = 6 corresponds to 2006, we must find P(6).

P(x) = 0.808x2 + 2.625x + 0.502

P(6) = 0.808(6)2 + 2.625(6) + 0.502 Let x = 6.

= 45.34 Evaluate.

Thus, in 2006 about 45.34 million households are expected to pay at least

one bill on-line each month.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 9

10.
6.3 Polynomial Functions

Adding and Subtracting Functions

Adding and Subtracting Functions

If f(x) and g(x) define functions, then

(f + g) (x) = f (x) + g(x) Sum function

and (f – g) (x) = f (x) – g(x). Difference function

In each case, the domain of the new function is the intersection of the

domains of f(x) and g(x).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 10

Adding and Subtracting Functions

Adding and Subtracting Functions

If f(x) and g(x) define functions, then

(f + g) (x) = f (x) + g(x) Sum function

and (f – g) (x) = f (x) – g(x). Difference function

In each case, the domain of the new function is the intersection of the

domains of f(x) and g(x).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 10

11.
6.3 Polynomial Functions

EXAMPLE 3 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 2x2 – 3x + 4 and g(x) = x2 + 9x – 5,

find (a) the sum and (b) the difference.

(a) (f + g) (x) = f (x) + g(x) Use the definition.

= (2x2 – 3x + 4) + (x2 + 9x – 5)Substitute.

= 3x2 + 6x – 1 Add the polynomials.

(b) (f – g) (x) = f (x) – g(x) Use the definition

= (2x2 – 3x + 4) – (x2 + 9x – 5) Substitute.

= (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtract

to addition.

= x2 – 12x + 9 Add.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 11

EXAMPLE 3 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 2x2 – 3x + 4 and g(x) = x2 + 9x – 5,

find (a) the sum and (b) the difference.

(a) (f + g) (x) = f (x) + g(x) Use the definition.

= (2x2 – 3x + 4) + (x2 + 9x – 5)Substitute.

= 3x2 + 6x – 1 Add the polynomials.

(b) (f – g) (x) = f (x) – g(x) Use the definition

= (2x2 – 3x + 4) – (x2 + 9x – 5) Substitute.

= (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtract

to addition.

= x2 – 12x + 9 Add.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 11

12.
6.3 Polynomial Functions

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(a) (f + g) (5)

(f + g) (5) = f (5) + g(5) Use the definition.

= [4(5)2 – 5] + 3(5) Substitute.

= 110

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 12

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(a) (f + g) (5)

(f + g) (5) = f (5) + g(5) Use the definition.

= [4(5)2 – 5] + 3(5) Substitute.

= 110

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 12

13.
6.3 Polynomial Functions

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(a) (f + g) (5)

Alternatively, we could first find (f + g) (x).

(f + g) (x) = f (x) + g(x) Use the definition.

= (4x2 – x) + 3x Substitute.

= 4x2 + 2x

(f + g) (5) = 4(5)2 + 2(5) = 110. The result is the same.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 13

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(a) (f + g) (5)

Alternatively, we could first find (f + g) (x).

(f + g) (x) = f (x) + g(x) Use the definition.

= (4x2 – x) + 3x Substitute.

= 4x2 + 2x

(f + g) (5) = 4(5)2 + 2(5) = 110. The result is the same.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 13

14.
6.3 Polynomial Functions

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(b) (f – g) (x) and (f – g) (3)

(f – g) (x) = f (x) – g(x) Use the definition.

= (4x2 – x) – 3x Substitute.

= 4x2 – 4x Combine like terms.

(f – g) (3) = 4(3)2 – 4(3) = 24. Substitute.

Confirm that f (3) – g(3) gives the same result.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 14

EXAMPLE 4 Adding and Subtracting Functions

For the polynomial functions defined by

f(x) = 4x2 – x and g(x) = 3x,

find each of the following.

(b) (f – g) (x) and (f – g) (3)

(f – g) (x) = f (x) – g(x) Use the definition.

= (4x2 – x) – 3x Substitute.

= 4x2 – 4x Combine like terms.

(f – g) (3) = 4(3)2 – 4(3) = 24. Substitute.

Confirm that f (3) – g(3) gives the same result.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 14

15.
6.3 Polynomial Functions

Basic Polynomial Functions

The simplest polynomial function is the identity function, defined by f(x) = x.

y

x f(x) = x

–2 –2

–1 –1

0 0 x

1 1

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 15

Basic Polynomial Functions

The simplest polynomial function is the identity function, defined by f(x) = x.

y

x f(x) = x

–2 –2

–1 –1

0 0 x

1 1

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 15

16.
6.3 Polynomial Functions

Basic Polynomial Functions

The squaring function, is defined by f(x) = x2.

y

x f(x) = x2

–2 4

–1 1

0 0 x

1 1

2 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 16

Basic Polynomial Functions

The squaring function, is defined by f(x) = x2.

y

x f(x) = x2

–2 4

–1 1

0 0 x

1 1

2 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 16

17.
6.3 Polynomial Functions

Basic Polynomial Functions

The cubing function, is defined by f(x) = x3.

y

x f(x) = x3

–2 –8

–1 –1

0 0 x

1 1

2 8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 17

Basic Polynomial Functions

The cubing function, is defined by f(x) = x3.

y

x f(x) = x3

–2 –8

–1 –1

0 0 x

1 1

2 8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 17

18.
6.3 Polynomial Functions

EXAMPLE 5 Graphing Variations of the Identity Function

Graph the function by creating a table of ordered pairs. Give the domain and

the range of the function by observing the graph.

(a) f(x) = –2x. y

Range

x f(x) = –2x

–2 4

–1 2

Domain x

0 0

1 –2

2 –4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 18

EXAMPLE 5 Graphing Variations of the Identity Function

Graph the function by creating a table of ordered pairs. Give the domain and

the range of the function by observing the graph.

(a) f(x) = –2x. y

Range

x f(x) = –2x

–2 4

–1 2

Domain x

0 0

1 –2

2 –4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 18

19.
6.3 Polynomial Functions

EXAMPLE 5 Graphing Variations of the Identity Function

Graph the function by creating a table of ordered pairs. Give the domain and

the range of the function by observing the graph.

y

(b) f(x) = x2 – 2.

2 Range

x f(x) = x – 2

–2 2

–1 –1

Domain x

0 –2

1 –1

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 19

EXAMPLE 5 Graphing Variations of the Identity Function

Graph the function by creating a table of ordered pairs. Give the domain and

the range of the function by observing the graph.

y

(b) f(x) = x2 – 2.

2 Range

x f(x) = x – 2

–2 2

–1 –1

Domain x

0 –2

1 –1

2 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 19