We are going to put all of our knowledge together for finding Zeros, either by factoring or Synthetic Division, and learn how to Graph a Polynomial Function!
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
The steps or guidelines for Graphing Polynomial Functions are very straightforward, and helps to organize our thought process and ensure that we have an accurate graph.
We will
- First find our y-intercepts and use our Number of Zeros Theorem to determine turning points and End Behavior patterns.
- Next we will find our Zeros (roots) by either factoring or the Rational Zeros Theorem (i.e., synthetic division).
- Lastly, we will discover how to graph each polynomial by analyzing the type of zeros we obtain. This means we will learn how to handle single (simple) zeros, even multiplicity, and odd multiplicity.
What’s important to note is that every polynomial function is smooth and continuous, and as Paul’s Online Notes so accurately states, this means that there are no holes or breaks or sharp corners in the graph.
Identifying Maximum and Minimums
But, we won’t stop there!
For each polynomial that we graph, we will look ahead to calculus and take the time to discuss maximums and minimums (turning points), concavity (how the graph opens), increasing and decreasing intervals, and also limits at infinity (end behavior)!
Additionally, we will look at the Intermediate Value Theorem for Polynomials, also known as the Locator Theorem, which shows that a polynomial function has a real zero within an interval.
And, we will contrast this with the Intermediate Value Theorem for Functions, which shows how to prove that a function is continuous.
Graphing Polynomial Functions – Video
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