Graphing Polynomials - End Behavior

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Objective

SWBAT determine the end behavior of a polynomial function by examining it's algebraic structure.

Big Idea

A good sketch of a polynomial function can be produced by considering the end-behavior, roots and y-intercept of a polynomial function.

Warm Up: Graphing Linear and Quadratic Functions

15 minutes

To begin thinking about the graphs of polynomial functions, my students will start the lesson by completing Warm-up Graphing Linear and Quadratic Functions.  In this quick warm up, students are presented with two linear and two quadratic functions to graph. I expect them to perform this Warm Up task quickly since the content is review.

Question Answering Session

40 minutes

I present a typed-up version of the questions generated in the previous day's session, which might look something like Summary of Polynomial Graph Questions. There are some questions in this list that need to be answered in this lesson. So, even if the students didn't come up with them yesterday, I will add them in. Specifically, I will make sure there is a question about end behavior and zeros of the function.  

Initially I give my students some time to formulate answers to these questions independently. After 10 minutes or so, I will ask for a volunteer to attempt to answer the first question (regarding end-behavior).  Working with the answer provided by the volunteer, I guide the students to build on each other's answers until we reach an answer that is thorough, but in the students' own words [MP3].

My role in this discussion is to shape it and make sure that it is headed in a productive direction. I try to validate all ideas, but also to make sure that good ideas are recognized by all as moving the conversation in a positive direction.

When each answer is complete, I will summarize it on the board, emphasizing formal notation where necessary [MP6].  For example for end behavior students must be able to explain that when the degree of a polynomial is odd  and the lead coefficient is positive,

as x→∞,y→∞ (up on the right)

as  x→-∞,y→-∞ (down on the left)

The formal notation tends to be the most challenging part of the end behavior discussion, so I continue to include the verbal translation every time I use infinity notation. 

Stand Up and Practice End Behavior

15 minutes

In order to solidify understanding of end behavior and give the students a chance to move around, we take 10 minutes to complete Stretch Break - Polynomial End Behavior.  I put on some music that my students like and slowly go through the slides, which have one function written on each slide.  The goal is for students to model the end behavior of each function with their arms.  If I pick the right music, I can count on high engagement in this activity, and it helps students solidify their understanding of end behavior [MP7].

Review Assignment

20 minutes

For this evening's homework, I ask my students to complete WS Review and Polynomial End Behavior, a 28-question packet that has some questions on polynomial end behavior and some review questions on polynomial skills and concepts we have learned previously.  The solutions to this packet will be available on Edmodo and I encourage my students to check their answers before they come to class.

I stop for review here because in the next lesson we will put our work with polynomial theorems together with the graphs of polynomial polynomial functions.  This will require my students to apply almost everything they have learned about polynomials.