Polynomial Functions and Models

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Presentation transcript:

Polynomial Functions and Models Section 4.1 Polynomial Functions and Models Copyright © 2013 Pearson Education, Inc. All rights reserved

Identify polynomial functions and their degree. Objectives Identify polynomial functions and their degree. Graph polynomial functions using transformations. Identify the real zeros of a polynomial function and their multiplicity. Analyze the graph of a polynomial function. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Properties of Power Functions f(x)=xn where n is an even integer: Symmetric with respect to y-axis. D: (‒∞,∞) and R: [0,∞) Always contains (-1,1), (0,0), and (1,1). As n increases, in magnitude, the graph increases more rapidly when x<-1 or x>1, but near the origin, the graph flattens. Copyright © 2013 Pearson Education, Inc. All rights reserved

Properties of Power Functions f(x)=xn where n is an odd integer: Symmetric with respect to origin. D and R: (‒∞,∞) Always contains (-1,‒1), (0,0), and (1,1). As n increases, in magnitude, the graph increases more rapidly when x<-1 or x>1, but near the origin, the graph flattens. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Find a polynomial of degree 3 whose zeros are -4, -2, and 3. Use a graphing utility to verify your result. Copyright © 2013 Pearson Education, Inc. All rights reserved

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. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Turning Points (changes from increasing ↔ decreasing): If f is a polynomial function of degree n, f has at most n‒1 turning points. If the graph of a polynomial function f has n‒1 turning points, the degree of f is at least n. Copyright © 2013 Pearson Education, Inc. All rights reserved

y = 4(x - 2) Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Use this notation: Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

The polynomial is degree 3 so the graph can turn at most 2 times. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Homework 4.1: # 13, 14, 15, 19, 41, 47, 49, 57, 61, 65, 69, 73 (#69 and 73: steps 1-3,6,7)