Polynomials in the Long Run (Lesson 2.2 Day 2)
Unit 2 - Day 4
Identify the degree and leading coefficient of any polynomial function
Describe a polynomial’s end behavior by looking at its degree and leading coefficient
Sketch the graph of a polynomial function by attending to its x- and y-intercepts, zeros, and end behavior
Quick Lesson Plan
For this lesson, we really want students to understand “end behavior” in an algebraic sense to prepare them for future lessons in Precalculus and Calculus involving asymptotes and limits. They can choose any values to test for questions 1 and 2, but make sure they follow the “as x gets bigger and bigger” and “more and more negative” and choose large values to test so they can make the connection between positive and negative infinity.
The idea is for them to explore the relationship between the exponents, leading coefficients, and “behavior” at each end of the graph on their own so they will understand the more technical way to describe the end behavior of a polynomial. Questions 3-5 are an attempt to show that only the leading term matters when determining end behavior as well as circling back to the transformations we learned in Unit 1.
Later in this unit, we will ask students to find horizontal asymptotes from rational functions, so the idea of plugging in extreme values at each end of the graph will help to make the connections there as well.
We build up to the official notation by starting with “left and right side” of the graph, moving to “as x gets…” and then finally “.” The main takeaway from the experience is the table in the Important Ideas box. They should be able to see a polynomial function and determine its end behavior without graphing. Polynomials of even degree have the same end behavior on the left and right side of the graph due to the nature of the even exponent. Polynomials of odd degree have opposite end behavior on the left and right side due to the nature of the odd exponent. The leading coefficient affects the overall direction of the polynomial.