Lesson 18

Graphs of Rational Functions (Part 2)

18.1: Rewritten Equations (5 minutes)

Warm-up

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.

Student Facing

Decide if each of these equations is true or false for \(x\) values that do not result in a denominator of 0. Be prepared to explain your reasoning.

  1. \(\displaystyle{\frac{x+7}{x}=1+\frac{7}{x}}\)
  2. \(\displaystyle{\frac{x}{x+7} =1+\frac{x}{7}}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

18.2: Publishing a Paperback (15 minutes)

Activity

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.

Launch

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.

Conversing, Writing: MLR2 Collect and Display. Before students begin writing a response to the last question, invite them to discuss their thinking with a partner. Listen for and collect vocabulary, gestures, and phrases students use to describe the function’s behavior as it approaches infinity. Capture student language that reflects a variety of ways to describe the behavior of the function near the non-zero horizontal asymptote. Write the students’ words on a visual display or add it to the display from the previous lesson and update it throughout the remainder of the lesson, with phrases such as “gets close to 4,” “never reaches,” and “almost flat.” Add the term “horizontal asymptote” to the chart when introduced in the activity synthesis. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. When students have seen that \(\frac{4x+120}{x}\) can also be expressed as \(4+\frac{120}{x}\), they can sketch the graph, record both expressions on a display nearby, annotate to show the horizontal asymptote and end behavior, and record the terms in the expressions associated with each. Students can use this as a reference in the next activity.
Supports accessibility for: Organization; Conceptual processing; Memory

Student Facing

Let \(c\) be the function that gives the average cost per book \(c(x)\), in dollars, when using an online store to print \(x\) copies of a self-published paperback book. Here is a graph of \(c(x)= \tfrac{120+4x}{x}.\)

Coordinate plane, total books printed, 0 to 500 by 50, cost per book, 0 to 20 by 1. Curve beings near 1 comma 20 and drops steeply toward 50 comma 6 point 2, then gently toward 500 comma 4 point 1.
  1. What is the approximate cost per book when 50 books are printed? 100 books?
  2. The author plans to charge \$8 per book. About how many should be printed to make a profit?
  3. What is the value of \(c(x)\) when \(x=\frac{1}{2}\)? How does this relate to the context?
  4. What does the end behavior of the function say about the context?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(x\).

Activity Synthesis

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.

18.3: Horizontal Asymptotes (15 minutes)

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.

Launch

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.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses to the question “Where do you see the horizontal asymptote of the graph in the expressions for the functions?” Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, ”Can you give an example?”, “How did _____ help you?”, “First, I _____ because . . . .”, and “Is it always true that . . . ?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students clarify their reasoning and understanding of rational functions with horizontal asymptotes.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “I matched ____ to ____ , because the asymptote was . . . .”, “I noticed . . . in the graph matched _____ in the equation, so I . . . .”, “Why did you . . . ?”, and “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Here are four graphs of rational functions.

A

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

B

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

C

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

D

Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.

​​​​​

  1. Match each function with its graphical representation.
    1. \(a(x)=\frac{4}{x}-1\)
    2. \(b(x)=\frac{1}{x}-4\)
    3. \(c(x)=\frac{1+4x}{x}\)
    4. \(d(x)=\frac{x+4}{x}\)
    5. \(e(x)=\frac{1-4x}{x}\)
    6. \(f(x)=\frac{4-x}{x}\)
    7. \(g(x)=1+\frac{4}{x}\)
    8. \(h(x)=\frac{1}{x}+4\)
  2. Where do you see the horizontal asymptote of the graph in the expressions for the functions?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Consider the function \(a(x) = \frac{\frac{1}{2}x+1}{x-1}\).

  1. Predict where you think the vertical and horizontal asymptotes of \(a(x)\) will be. Explain your reasoning.
  2. Use graphing technology to check your prediction.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

The purpose of this math talk is to make sure students can identify the horizontal asymptote of a rational function from the equation and to get students wondering about how to figure out horizontal asymptotes when the denominator isn’t just \(x\).

Display the following functions one at a time for all to see. Give students quiet think time for each function and ask them to give a signal when they have an answer and a strategy. Invite students to share their strategy, recording and displaying responses for all to see, before revealing the next function. Keep all functions displayed throughout the talk.

Identify the horizontal asymptote of the function.

  • \(f(x)=\frac{7x+2}{x}\)
  • \(g(x)=\frac{7x+2}{2x}\)
  • \(h(x)=\frac{7x+2}{0.5x}\)

(\(f\): \(y=7\), \(g\): \(y=3.5\), \(h\): \(y=14\))

Here are some questions for discussion:

  • “What are two ways you could change a single number in \(f(x)\) so it has a horizontal asymptote at 35?” (Change the 7 to a 35, or change the coefficient of the \(x\) in the denominator to \(\frac15\).)
  • “What is the vertical asymptote for each function?” (Each has a vertical asymptote at \(x=0\).)
  • “How could you figure out the horizontal asymptote of something like \(k(x)=\frac{7x+2}{x-1}\)?” (We could graph the function and zoom out to see what the value \(k(x)\) gets closer to for large values of \(x\).)

18.4: Cool-down - Publishing a Paperback, Revisited (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Consider the rational function \(f(x) = \frac{3x+1}{x}\). Written this way, we can tell that the graph of the function has a vertical asymptote at \(x=0\) by reading the denominator and identifying the value that would cause division by zero. But what can we tell about the value of \(f(x)\) for values of \(x\) far away from the vertical asymptote?

One way we can think about these values is to rewrite the expression for \(f(x)\) by breaking up the fraction:

\(f(x) = \frac{3x}{x} + \frac{1}{x} \\ f(x)= 3 + \frac{1}{x}\)

Written this way, it’s easier to see that as \(x\) gets larger and larger in either the positive or negative direction, the \(\frac{1}{x}\) term will get closer and closer to 0. Because of this, we can say that the value of the function will get closer and closer to 3.

More generally, if a rational function \(g(x) = \frac{a(x)}{b(x)}\) can be rewritten as \(g(x) = c + \frac{r(x)}{b(x)}\), where \(c\) is a constant, and \(r(x)\) and \(b(x)\) are polynomial expressions where \(\frac{r(x)}{b(x)}\) gets closer and closer to zero as \(x\) gets larger and larger in both the positive and negative directions, then \(g(x)\) will get closer and closer to \(c\).

Rational functions of this type have a horizontal asymptote at the constant value. The line \(y=c\)  is a horizontal asymptote for \(f\) if \(f(x)\) gets closer and closer to \(c\) as the magnitude of \(x\) increases.