LEARNING OBJECTIVES
By the end of this lesson, you will be able to:
- Recognize characteristics of graphs of polynomial functions.
- Use factoring to find zeros of polynomial functions.
- Identify zeros and their multiplicities.
- Determine end behavior.
- Understand the relationship between degree and turning points.
- Graph polynomial functions.
- Use the Intermediate Value Theorem.
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.