Graphing Polynomial Functions

Theorem (Graphing Polynomial Functions) Leading Coefficient Test. As moves without bound to the left or to the right, the graph of the polynomial function



eventually rises or falls according to:

    (i) If is odd and then the graph falls to the left and rises to the right.
    
    (ii) If is odd and then the graph falls to the right and rises to the left.

    (iii) If is even and then the graph falls to the right and right.

    (iv) If is even and then the graph rises to the right and left.

Example (Graphing Polynomial Functions) Leading Coefficient Test. Determine the long term behavior of the polynomial functions by using the Leading Coefficient Test.

    (a) Consider the polynomial function since is even and the graph of rises to the right as and rises to the left as ; as shown in the following graph of



    (b) Consider the polynomial function since is odd and is the leading coefficient the graph of falls to the right as and rises to the left as ; as shown in the following graph of

    Even though the tools of calculus may not be available a decent graph of a polynomial can be made by factoring the polynomial and looking at intercepts and long term behavior of the graph of the polynomial.  

Example (Graphing Polynomial Functions) The Factoring Method. To graph we begin by factoring



Its only roots are and So has no roots in any of these intervals: and Using the Intermediate Value Theorem it suffices to test each of these intervals by choosing a interval representative and testing if is positive or negative. We summarize our results in the following table:



This information can be used to make a rough  sketch of the polynomial function.

Example (Graphing Polynomial Functions) The Factoring Method. To graph we begin by factoring



Its only roots are and So has no roots in any of these intervals: and Using the Intermediate Value Theorem it suffices to test each of these intervals by choosing a interval representative and testing if is positive or negative. We summarize our results in the following table:



This information can be used to make a rough  sketch of the polynomial function.

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Graphing Polynomial Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/graphing-polynomial-functions.html