Section 4.1 Polynomial Functions and Models. Copyright 2013 Pearson Education, Inc. All rights reserved
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1 Section 4.1 Polynomial Functions and Models Copyright 2013 Pearson Education, Inc. All rights reserved
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4 3 8 ( ) = + (a) f x 3x 4x x (b) ( ) g x 2 x + 3 = x 1 (a) f is a polynomial of degree 8. (b) g is not a polynomial function. It is the ratio of two distinct polynomials. ( ) (c) h x = 5 (c) h is a polynomial function of degree 0. 0 ( ) x ( ) (d) F x = ( x 3)( x+ 2) 2 It can be written h x = 5 = 5. It can be written F( x) = x x ( ) = (f) ( ) (e) G x 3x 4x (e) G is not a polynomial function. The second term does not have a nonnegative integer exponent. (d) F is a polynomial function of degree H x = x x + x (f) H is a polynomial of degree 3. Copyright 2013 Pearson Education, Inc. All rights reserved
5 Summary of the Properties of the Graphs of Polynomial Functions Copyright 2013 Pearson Education, Inc. All rights reserved
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20 Find a polynomial of degree 3 whose zeros are -4, -2, and 3. Use a graphing utility to verify your result. f x = a x+ 4 x+ 2 x 3 ( ) ( )( )( ) ( ) = ( + 4)( + 2)( 3) f x x x x ( ) = a x + x x ( ) = 2( + 4)( + 2)( 3) f x x x x ( ) = ( + 4)( + 2)( 3) f x x x x Copyright 2013 Pearson Education, Inc. All rights reserved
21 For the polynomial, list all zeros and their multiplicities. 3 4 f x = 2 x 2 x+ 1 x 3 ( ) ( )( ) ( ) 2 is a zero of multiplicity 2 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. Copyright 2013 Pearson Education, Inc. All rights reserved
22 ( ) = ( 3) 2 f x x x ( ) ( ) 2 2 (a) x-intercepts: 0 = x x 3 x= 0 or x 3 = 0 x= 0 or x= 3 y-intercept: f ( 0) = 0( 0 3) 2 = 0 y = 0 Copyright 2013 Pearson Education, Inc. All rights reserved
23 ( ) = ( 3) 2 f x x x ( 0,0 ),( 3,0) (,0) ( 0,3 ) ( 3, ) 1 f ( 1) = 16 Below x-axis ( 1, 16) 1 f ( 1) = 4 Above x-axis ( 1, 4) 4 f ( 4) = 4 Above x-axis ( 4, 4) Copyright 2013 Pearson Education, Inc. All rights reserved
24 y ( ) = ( 3) 2 f x x x x (,0) ( 0,3 ) ( 3, ) 1 f ( 1) = 16 Below x-axis ( 1, 16) 1 f ( 1) = 4 Above x-axis ( 1, 4) 4 f ( 4) = 4 Above x-axis ( 4, 4) Copyright 2013 Pearson Education, Inc. All rights reserved
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27 y = 4(x - 2) Copyright 2013 Pearson Education, Inc. All rights reserved
28 y = 4(x - 2) Copyright 2013 Pearson Education, Inc. All rights reserved
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39 f ( ) 0 = 6 so the y intercept is 6. The degree is 4 so the graph can turn at most 3 times. 4 For large values of x, end behavior is like x (both ends approach ) Copyright 2013 Pearson Education, Inc. All rights reserved
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43 1 The zero has multiplicity 1 2 so the graph crosses there. The zero 3 has multiplicity 2 so the graph touches there. Copyright 2013 Pearson Education, Inc. All rights reserved
44 The polynomial is degree 3 so the graph can turn at most 2 times. Copyright 2013 Pearson Education, Inc. All rights reserved
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52 The domain and the range of f are the set of all real numbers. ( ) ( ) ( ) Decreasing: 2.28, 0.63 Increasing:, 2.28 and 0.63, Copyright 2013 Pearson Education, Inc. All rights reserved
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56 A cubic relation may exist between the two variables. Copyright 2013 Pearson Education, Inc. All rights reserved
57 Cubic function of best fit: Copyright 2013 Pearson Education, Inc. All rights reserved
58 Cubic function of best fit: Copyright 2013 Pearson Education, Inc. All rights reserved
Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics
Section 5.1 Polynomial Functions and Models
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5 Exponents, Polynomials, and Polynomial Functions Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1 5.1 Integer Exponents R.1 Fractions and Scientific Notation Objectives 1. Use the product
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