Graphs of Polynomial Functions
Lesson 6 of 15
Objective: Describe the end behavior of and sketch graphs of polynomial functions.
Warm up and Homework Review
I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogenous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains the lesson's Warm Up- Graphs of Polynomial Functions which uses transformations of functions to discuss the end behavior of a simple parabola. This connects to the discussion of end behavior later in the lesson.
I also use this time to correct and record the previous day's Homework.
We start with the parent functions: f(x) = x2 to f(x) = x7. My students have already been exposed to the concept of parent functions for quadratic, linear, absolute value, and rational functions. I present each one and allow time for student to make quick sketches in their notes. I mark the vertex or center of each parent graph as well as the two points one unit way. Focusing on these points gives the students a pattern to follow as well as points to accurately show all transformations, including stretches/shrinks, in future lessons.
Once we have covered all six, the students talk with their partners about any patterns they see, followed by a group discussion. I make sure to ask WHY each observed pattern is true. Then the students record a statement describing the pattern(s) including the WHY in their notes.
This leads into a discussion on end behavior for even degree functions. First we define even functions. We do a think-pair-share on functions we have already seen that are even. Next we discover/discuss end behavior of even functions. I choose not to give a formal definition yet since I want my students to develop a solid conceptual understanding. I find that students often get stuck on formal definitions and never really grasp what is being said. A formal definition can always be introduced later once students master the concept.
I have students define with their partner what the graph ends are "doing" and then share out as a whole class. Definitions should include [increasing or decreasing] behaviors for BOTH extremes of the graphs. I encourage the students to evaluate their definitions for completeness and accuracy (Math Practice 3). To wrap up the conversation, students write down their definitions including diagrams that may be helpful.
I then repeat the conversation now for the opposite [negative] even functions and again for odd functions. Each discussion should move along faster since students are building on their initial concept of end behavior.
End Behavior Aerobics
Now that students have looked the end behavior of parent even and odd functions, I give them the opportunity to determine end behavior of more complex polynomials. The objective is that the students make the connection that the degree of a polynomial affects the graph's end behavior. The students look for end behavior patterns by entering five polynomial equations into their graphing calculator (Math Practice 7).
I give the students the task and then walk around the classroom giving feedback as necessary. A useful classroom technique is to walk through the classroom giving lots of positive feedback on correct descriptions. Students will do a lot to receive recognition of work well done. Scaffolding can be provided to students during this time as well. For example, if a student seems stuck you can ask “What exponent [even/odd] has that type of end behavior? Do we have that of exponent in our polynomial?”
It is important to be mindful with pacing. This activity should only take a couple of minutes. Once many students seem to have a good idea how to identify the end behavior, I ask for a volunteer to explain how to find the end behavior from an equation.
Now, we do some end behavior aerobics. Students stand up and show with their hands what each type of behavior looks like. I model each movement with my arms to get them started. Enthusiasm on my part helps to persuade some of the reluctant students. I also let them know that modeling things physically with your body increases the likelihood of remembering it. I got the idea for this activity from my fellow teacher David Winkelman.
Please see the PowerPoint for detailed presentation notes.
The final portion of this lesson has students sketch the graphs of polynomials. I start this section with a list of my expectations for an acceptable polynomial graph. They are:
- The zeros of the polynomial
- The y-intercept
- Appropriate end behavior
- One point between each zero to approximate the shape
I have the students write this down and put a box around it to stress its importance. For each graph, they should include these features. The y-intercept can count as one of the approximation points. For the other approximations points, I encourage them estimate the middle of each interval.
This Homework starts with 6 polynomials where students describe the end behavior. The next four, students sketch a graph of each polynomial. The goal of this assignment is to reinforce the day's lesson.
This assignment was created with Kuta Software, an amazing resource for secondary mathematics teachers.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
This Exit Ticket checks to see how many students can describe the end behavior of a polynomial.