Lesson 18

The purpose of this task is for students to consider a common mistake when rewriting rational expressions. In the following activities, students will rewrite rational expressions to reveal end behavior using their understanding of fractions, and in the next lesson, polynomial long division.

Select 2–3 students to share if they think an equation is true or false and why. If any students tried to use long division to make sense of \(\frac{x}{x+7} =1+\frac{x}{7}\), invite them to share their thinking. Students will do more work with long division and rational expressions in the next lesson, however, so this is not a technique that needs to be examined in depth at this time.

This activity is the first time students study a rational function with a non-zero horizontal asymptote. The first part of the activity is for students to make sense of the context while reasoning about input-output pairs. The questions in the activity and the following discussion ask students to reason about why the cost per book trends toward $4 from first a context perspective, and then by rewriting the expression for the function and interpreting it within the context.

Monitor for students who have clear explanations regarding what the end behavior says about the context.

Display the opening statement and graph for all to see. Give students 1 minute to read the statement and then ask, “In the expression for \(c(x)\), there is division by \(x\) since it gives the cost per book. What do you think the \(120+4x\) means?” (The \(120+4x\) is the cost to make \(x\) books, which is $4 per book and a $120 set-up fee.) Once students understand the different parts of the expression, allow them to continue with the first question in the activity.

If time allows, encourage students to calculate more accurate responses to the first three questions by using the equation of the function. Otherwise, approximating using the graph is the expectation.

Students who are only looking at the graph might think that the end behavior is close to $0. Suggest substituting a large number for \(x\) or looking at a table to see the value of \(c(x)\) for large values of \(x\).

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.

Begin the discussion by selecting students to share how they worked out solutions to the first 3 questions. If any students used the equation to answer these questions exactly and time allows, invite them to share their process, particularly for the question asking how many books must be printed in order to make a profit. Students will have more opportunities to solve rational equations algebraically in future lessons, so this does not need to be a focus at this time.

Select previously identified students to share what the end behavior of the function says about the context. Then, ask students how we could rewrite \(\frac{120+4x}{x}\) to make the end behavior more obvious when looking at the expression for \(c\). After quiet work time, invite students to share their ideas, recording these for all to see. Hopefully, at least one student has written \(4+\frac{120}{x}\), 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 \(4+\frac{120}{x}\) uncovers the end behavior?” (The 4 uncovers the end behavior of \(c\) because when \(x\) gets larger and larger in the positive direction, \(\frac{120}{x}\) gets closer to 0.) Tell students that this type of end behavior is due to a horizontal asymptote. For large numbers of books printed, the value of \(c(x)\) will get closer and closer to $4 but never quite reach it 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 (just think back to the graphs of the cylinder), but those that do can be written a certain way, which is the focus of the next activity.

Building on the work of the previous activity, now students match graphs and equations of rational functions without context. The goal of this activity is for students to identify key features of graphs and equations, as well as practice rewriting rational expressions in different forms.

Monitor for students who rewrite the expressions in a different form to help reason about the matches.

Since this activity was designed for students to answer by reasoning about the expressions, students should not use graphing technology to identify matches.

Tell students to close their books or devices. Display the 4 graphs from the activity for all to see and ask students to identify the horizontal asymptote of each graph. After a brief quiet think time, select students to share what horizontal asymptotes they identified and record these next to the appropriate graph in the form \(y=\) . Allow students to reopen their books or devices and continue with the 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 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.