Students use the structure of expressions to match equations and graphs of polynomials (MP7). The details of what each graph looks like close to an input of 0 is concealed so that students must rely on their knowledge of end behavior to make the matches. While three of the graphs show end behavior students have encountered in previous lessons, one of the graphs has end behavior opposite of what they have seen. This graph matches an equation with a negative leading coefficient. The connection between a negative leading coefficient and “flipped” end behavior is explored further in the following activity, so only an informal understanding is expected at this time.

Monitor for students using different strategies to identify a matching equation for the third graph. For example, some students may test different inputs, while others may rewrite as and reason about the effect of the -1 coefficient on the leading term.

In this activity, students are using structure to reason about the matches, so technology is not an appropriate tool. 

The purpose of this discussion is to consider end behavior that has not been seen before and how it might be expressed in an equation. Some questions to start the discussion:

• “Which graphs must match even-degree polynomials? Odd-degree polynomials?” (Graphs A and B are even degree since the end behavior on each side matches. Graphs C and D must be odd-degree polynomials since the end behavior on each side is opposite.)
• “How did you identify the match for Graph B?” (I rewrote each equation in standard form and saw that is the same as , so that must be the match to Graph B.)

Then select 2–3 previously identified students to share how they identified which equation matched the third graph, starting with students who used specific input-output pairs. Highlight any students who rewrote in standard form to see that it is a third degree polynomial. Tell students that in the next activity, they will investigate which parts of an equation do and do not affect the end behavior.

Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.

Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves as math learners.

This activity addresses how the sign of the leading coefficient changes the end behavior of a polynomial. Working in small groups is meant to give students more equations to compare as they learn which features of an equation match different output behaviors.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Monitor for students who write these different types of equations for the last question:

• Equations written in standard form
• Equations written in factored form
• Equations that are not written in standard form or factored form
• Equations where the leading coefficient is not the first term
• Equations with different odd degrees
• Equations with different numbers of terms

While we describe end behavior of different functions in similar ways, not all end behavior is the same. In this activity, students investigate two functions with different degrees but the same end behavior when , and they are asked to decide which function they believe is greater. Students can use graphs, expressions, or tables to analyze the functions. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

A higher-degree function will always have output values that eventually exceed in magnitude the outputs of a lower-degree function. Both functions in this activity increase in the negative direction as increases in the positive direction, so the lower-degree function will have greater values. Students will need to be clear about their understanding of “greater” when explaining their thinking to the class (MP3).

Monitor for students using different strategies to build their case for why they think one function is greater than another. For example, some students may graph each function on the same axes, while others may make a table of various input-output pairs.