Copyright © 2011 Pearson, Inc. 2.3 Polynomial Functions of Higher Degree with Modeling.

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Presentation transcript:

Copyright © 2011 Pearson, Inc. 2.3 Polynomial Functions of Higher Degree with Modeling

Copyright © 2011 Pearson, Inc. Slide What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be used to provide approximations to more complicated functions, as you will see if you study calculus.

Copyright © 2011 Pearson, Inc. Slide The Vocabulary of Polynomials

Copyright © 2011 Pearson, Inc. Slide Example Graphing Transformations of Monomial Functions

Copyright © 2011 Pearson, Inc. Slide Example Graphing Transformations of Monomial Functions

Copyright © 2011 Pearson, Inc. Slide Cubic Functions

Copyright © 2011 Pearson, Inc. Slide Quartic Function

Copyright © 2011 Pearson, Inc. Slide Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

Copyright © 2011 Pearson, Inc. Slide Leading Term Test for Polynomial End Behavior

Copyright © 2011 Pearson, Inc. Slide Leading Term Test for Polynomial End Behavior

Copyright © 2011 Pearson, Inc. Slide Leading Term Test for Polynomial End Behavior

Copyright © 2011 Pearson, Inc. Slide Example Applying Polynomial Theory

Copyright © 2011 Pearson, Inc. Slide Example Applying Polynomial Theory

Copyright © 2011 Pearson, Inc. Slide Example Finding the Zeros of a Polynomial Function

Copyright © 2011 Pearson, Inc. Slide Example Finding the Zeros of a Polynomial Function

Copyright © 2011 Pearson, Inc. Slide Multiplicity of a Zero of a Polynomial Function

Copyright © 2011 Pearson, Inc. Slide Zeros of Odd and Even Multiplicity If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis at (c, 0) and the value of f changes sign at x = c. If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the x-axis at (c, 0) and the value of f does not change sign at x = c.

Copyright © 2011 Pearson, Inc. Slide Example Sketching the Graph of a Factored Polynomial

Copyright © 2011 Pearson, Inc. Slide Example Sketching the Graph of a Factored Polynomial

Copyright © 2011 Pearson, Inc. Slide Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y 0 is between f(a) and f(b), then y 0 =f(c) for some number c in [a,b]. In particular, if f(a) and f(b) have opposite signs (i.e., one is negative and the other is positive, then f(c) = 0 for some number c in [a, b].

Copyright © 2011 Pearson, Inc. Slide Intermediate Value Theorem

Copyright © 2011 Pearson, Inc. Slide Quick Review

Copyright © 2011 Pearson, Inc. Slide Quick Review Solutions