Polynomial Functions

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Sharp Tutor Mon, Jan 17, 2022 06:41 PM UTC

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. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 6.3 - 1

2. Chapter 6
Exponents, Polynomials, and
Polynomial Functions
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 6.3 - 2

3. 6.3
Polynomial Functions
Copyright © 2010 Pearson Education, Inc. All rights reserved
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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