Introduction to Graphs of Polynomial Functions

Learning Objectives

By the end of this lesson, you will be able to:

  • Identify zeros of polynomial functions with even and odd multiplicity
  • Use the degree of a polynomial to determine the number of turning points of its graph
  • Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the intermediate value theorem
  • Write the equation of a polynomial function given it’s graph

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below.

Year 2006 2007 2008 2009 2010 2011 2012 2013
Revenues 52.4 52.8 51.2 49.5 48.6 48.6 48.7 47.1

The revenue can be modeled by the polynomial function

[latex]R\left(t\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[/latex]

where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.