If students only look at the graph and think that the end behavior is close to \$0, consider asking:

• “Can you explain how you determined the end behavior of the function.”
• “How could substituting a large number for help you find the end behavior?”

The purpose of this discussion is for students to learn about and discuss what a horizontal asymptote is, building on what students already know about the end behavior of polynomial functions.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they solved the first 3 questions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Select previously identified students to share what the end behavior of the function says about the context. Then, ask students how we could rewrite to make the end behavior more obvious when looking at the expression for . After quiet work time, invite students to share their ideas, and record these for all to see. Hopefully, at least one student has rewritten the expression as , but if not, help guide students to this form by suggesting decomposing the single fraction into two fractions.

Once students understand how the new form was written, ask, “Which term in uncovers the end behavior?” (The 4 uncovers the end behavior of because, when gets larger and larger in the positive direction, gets closer to 0.) Tell students that this type of end behavior is due to a horizontal asymptote, and add this term to the reference chart. Tell students that a horizontal asymptote of a function is a line that the outputs of the function get closer and close to as the inputs get larger and larger in either the positive or negative direction. In this case, for large numbers of books printed, the value of will get closer and closer to but never quite reach \$4 due to the \$120 set-up fee that is divided among the total number of books printed.

Conclude the discussion by making sure students understand that not all rational functions have horizontal asymptotes (if needed, display the graphs of the cylinder from an earlier lesson), but those that do can be written a certain way, which is the focus of the next activity.

The goal of this discussion is for students to understand how their classmates reasoned about the different matches and to share their own reasoning.

Begin the discussion by asking previously identified students to share how they rewrote an expression and how doing so helped them identify which graph to match it with (or to say how rewriting didn’t help and what they did instead to identify the match). Encourage students to ask clarifying questions about the reasoning of their classmates. After each student shares, ask if any students identified the matching graph in a different way and invite those students to share their steps.