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Section 4.1 polynomial functions and models
1.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. Section 4.1 Polynomial Functions and Models
2.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
3.
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Pearson Education, Inc. Publishing as Prentice Hall.
4.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ) 3 8 (a) 3 4f x x x x= − + ( )(c) 5h x = ( ) 2 3 (b) 1 x g x x + = − ( )(d) ( 3)( 2)F x x x= − + ( ) 1 (e) 3 4G x x x− = − ( ) 3 21 2 1 (f) 2 3 4 H x x x x= − + (a) is a polynomial of degree 8.f (b) is not a polynomial function. It is the ratio of two distinct polynomials. g ( ) 0 (c) is a polynomial function of degree 0. It can be written 5 5. h h x x= = 2 (d) is a polynomial function of degree 2. It can be written ( ) 6. F F x x x= − − (e) is not a polynomial function. The second term does not have a nonnegative integer exponent. G (f) is a polynomial of degree 3.H
5.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. Summary of the Properties of the Graphs of Polynomial Functions
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Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
7.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
8.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
9.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
10.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
11.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. Find a polynomial of degree 3 whose zeros are -4, -2, and 3. Use a graphing utility to verify your result. ( ) ( ) ( ) ( )4 2 3f x a x x x= + + − ( ) ( ) ( ) ( )4 2 3f x x x x= + + − ( ) ( ) ( ) ( )4 2 3f x x x x= − + + − ( ) ( ) ( ) ( )2 4 2 3f x x x x= + + − ( )3 2 3 10 24a x x x= + − −
12.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ) ( ) ( ) ( ) 3 4 2 2 1 3f x x x x= − − + − For the polynomial, list all zeros and their multiplicities. 2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1. –1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3. 3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.
13.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ) ( ) 2 3f x x x= − ( ) ( ) 2 2 (a) -intercepts: 0 3 0 or 3 0x x x x x= − = − = 0 or 3x x= = ( ) ( ) 2 -intercept: 0 0 0 3 0y f = − = 0y =
14.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ) ( ) 2 3f x x x= − ( ) ( )0,0 , 3,0 ( ),0−∞ ( )0,3 ( )3,∞ 1− ( )1 16f − = − Below -axisx ( )1, 16− − 1 ( )1 4f = Above -axisx ( )1,4 4 ( )4 4f = Above -axisx ( )4,4
15.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ),0−∞ ( )0,3 ( )3,∞ 1− ( )1 16f − = − Below -axisx ( )1, 16− − 1 ( )1 4f = Above -axisx ( )1,4 4 ( )4 4f = Above -axisx ( )4,4 ( ) ( ) 2 3f x x x= − −3 −2 −1 1 2 3 4 5 6 −4 −3 −2 −1 1 2 3 4 x y
16.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
17.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
18.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. y = 4(x - 2)
19.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. y = 4(x - 2)
20.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( ) ( ) 2 3f x x x= −
21.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. Figure 16 (a)
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Pearson Education, Inc. Publishing as Prentice Hall. Figure 16 (b)
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Pearson Education, Inc. Publishing as Prentice Hall. Figure 16 (c)
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Pearson Education, Inc. Publishing as Prentice Hall. Figure 16 (d)
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Pearson Education, Inc. Publishing as Prentice Hall.
26.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
27.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. ( )0 6 so the intercept is 6.f y= − − The degree is 4 so the graph can turn at most 3 times. 4 For large values of , end behavior is like (both ends approach ).x x ∞
28.
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Pearson Education, Inc. Publishing as Prentice Hall.
29.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
30.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
31.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. 1 The zero has multiplicity 1 2 so the graph crosses there. − The zero 3 has multiplicity 2 so the graph touches there.
32.
Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall. The polynomial is degree 3 so the graph can turn at most 2 times.
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Pearson Education, Inc. Publishing as Prentice Hall.
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Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
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Copyright © 2012
Pearson Education, Inc. Publishing as Prentice Hall.
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