Lesson 17

Graphs of Rational Functions (Part 1)

17.1: Biking 10 Miles (Part 1) (5 minutes)

Warm-up

The purpose of this warm up is for students to recall and use the relationship between distance, rate, and time. Students build on their thinking here in the following activity.

Launch

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time followed by having students discuss responses with a partner, followed by a whole-class discussion.

Student Facing

A woman biking.

Kiran’s aunt plans to bike 10 miles.

  1. How long will it take if she bikes at an average rate of 8 miles per hour?
  2. How long will it take if she bikes at an average rate of \(r\) miles per hour?
  3. Kiran wants to join his aunt, but he only has 45 minutes to exercise. What will their average rate need to be for him to finish on time?
  4. What will their average rate need to be if they have \(t\) hours to exercise?

Student Response

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

Activity Synthesis

Select students to share their thinking for each question. If students did not include units with their answers, encourage them to do so now. While students may already be familiar with the different ways miles per hour is written, remind them that mph and \(\frac{\text{mi}}{\text{hr}}\) are both acceptable.

17.2: Biking 10 Miles (Part 2) (10 minutes)

Activity

In this activity, students write and then investigate a simple rational function relating distance, rate, and time for a bike ride. A simple context was chosen here in order to help students focus on what a vertical asymptote is and how to identify where one will be from an equation of a rational function (MP7). This activity is the beginning of the work students will do thinking about asymptotes, which continues into the next activity and the following lessons.

Launch

If an equation did not come up in the previous activity, ask students to create one now. Provide access to devices that can run Desmos or other graphing technology.

Conversing, Writing: MLR2 Collect and Display. Before students begin writing a response to the last two questions, 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 0. Capture student language that reflects a variety of ways to describe the behavior of the function near the vertical asymptote. Write the students’ words on a visual display and update it throughout the remainder of the lesson, including phrases such as “gets very large,” “never touches,” and “goes straight up.” Add the term “vertical 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

Student Facing

Kiran plans to bike 10 miles.

  1. Write an equation that gives his time \(t\), in hours, as a function of his rate \(r\), in miles per hour.
  2. Graph \(y=t(r)\).
  3. What is the meaning of \(t(8)\)? Does this value make sense? Explain your reasoning.
  4. What is the meaning of \(t(0)\)? Does this value make sense? Explain your reasoning.
  5. As \(r\) gets closer and closer to 0, what does the behavior of the function tell you about the situation?
  6. As \(r\) gets larger and larger, what does the end behavior of the function tell you about the situation?

Student Response

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

Anticipated Misconceptions

Some students may become distracted while graphing the function by the values when \(r\) is negative. Remind them to consider what an appropriate domain for the function is and to adjust their graphing windows appropriately.

Activity Synthesis

The purpose of this discussion is for students to learn about and discuss what a vertical asymptote is.

Begin the discussion by asking students what an acceptable domain for the function is. After a brief quiet think time, select students to share the domain they think is appropriate. If not brought up by students, ask, “Is 0 in the domain of the function?” (No, because you cannot divide by 0, and it doesn’t make sense that you could go 0 miles per hour and reach a distance 10 miles away.)

Display a graph of the function using an agreed-on domain to select an appropriate window size for all to see. Tell students that the way the graph curves up as \(r\) approaches 0 is a sign of a vertical asymptote. We can also tell that there is a vertical asymptote at \(r=0\) from the equation, because that is the input that leads to division by zero. As the value of \(r\) gets closer and closer to 0, the value of \(t\) gets greater and greater.

The last question asks students to make a connection between a previously learned idea (end behavior) and rational functions. Students will investigate horizontal asymptotes in the next lesson, so this question is meant only as a preview of the work to come and a chance for students to describe in their own words the end behavior of a function that is approaching a fixed value.

17.3: Card Sort: Graphs of Rational Functions (20 minutes)

Activity

This sorting task gives students opportunities to analyze equations and graphs closely to make connections between these two different representations of rational functions (MP7).

Monitor for different ways groups decide which function goes with which graph, but especially for those who use vertical intercepts and vertical asymptotes.

Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the following text and both graphs for all to see.

\(f\) and \(g\) are both rational functions defined by \(f(x)=\frac{6}{x}\) and \(g(x)=\frac{6}{x-1}\). Here are their graphs. What do you notice? What do you wonder?

Graph of \(g(x)= {6\over x-1}\).
Graph of \(f(x)=\frac{6}{x}\).

Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a brief whole-class discussion.

Things students may notice:

  • The graphs are shaped the same.
  • The graphs are mostly in two quadrants.
  • The end behavior is the same for both graphs.
  • Neither graph intersects the \(x\)-axis.
  • \(g\) has a vertical intercept at \((0,\text- 6)\).
  • \(g\) has a dashed vertical line at \(x=1\).

Things students may wonder:

  • Why does \(g\) have a dashed vertical line at \(x=1\)?
  • Would the graph of \(h(x)=\frac{6}{x-2}\) look like \(g(x)=\frac{6}{x-1}\) shifted 1 to the right?
  • What would the graph of \(j(x)=\frac{6}{x+1}\) look like?

Arrange students in groups of 2. Distribute pre-cut slips. Tell students that for any card with missing information, they will need to figure out what those cards match in order to complete the missing part.

Conversing: MLR8 Discussion Supports. In their groups of 2, students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ _____ and _____ match because . . . .”, and “I noticed _____ , so I matched . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about relationships between functions and the vertical asymptotes of their graphs.
Design Principle(s): Support sense-making; Maximize meta-awareness
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, students can highlight the asymptotes and annotate with the denominator that would result in an asymptote there.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Memory

Student Facing

Your teacher will give you a set of cards. Match each rational function with its graphical representation.

Student Response

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

Student Facing

Are you ready for more?

Priya and Han are bicycling. Han is going at a rate of 10 mph and begins 2 miles ahead of Priya. If Priya bikes at a rate of \(r\) mph, when will Priya pass Han? Write an equation and sketch a graph. Then interpret the graph in terms of the situation.

Student Response

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

Anticipated Misconceptions

Students who have trouble getting started may need to be reminded that they can use the same techniques they've used in the past to match graphs and functions: looking for horizontal and vertical intercepts, considering what happens as \(x\) gets larger in the positive or negative direction, and finding the function’s domain and range. To deal with asymptotes, they can recall the situation in the warm-up, in which the output of the function (time) got larger and larger as the input (speed) approached 0. In the previous activity, they saw how to tell by looking at the function for time that it would increase as the speed approached 0: this was the value of speed that would cause division by 0. Encourage them to apply this insight to the equations and graphs in the card sort.

Activity Synthesis

Select students previously identified to share how they matched the functions with the graphs. Choose as many different ways as time allows, but ensure that one group uses vertical intercepts and one group uses vertical asymptotes. Attend to the language that students use to describe their graphs, giving them opportunities to describe their graphs more precisely. Highlight the use of terms like vertical intercept, vertical asymptote, domain, range, increasing, decreasing, and end behavior.

If not mentioned by students, ask why some of the graphs appear “flipped” compared to others. This is an opportunity for students to recall what they learned about the effect of a negative leading coefficient with polynomial functions and apply that knowledge to rational functions.

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is to give students an opportunity to practice describing the graphs of rational functions with a focus on vertical asymptotes.

Arrange students in groups of 2. Tell the class that you are going to display several graphs of rational functions and that they should all choose one to focus on. Taking turns, each student will describe their chosen graph in words and their partner will try to guess which graph is theirs. Warn them that the graphs will not have any markings on the vertical axis, so students will have to focus on other features in their description. Students describing graphs should also not use the name of the graph as part of their description. Display the following graphs for all to see.

A

Graph of a rational function.

B

Graph of a rational function.

C

Graph of a rational function.

D

Graph of a rational function.

E

Graph of a rational function.

F

Graph of a rational function.

G

Graph of a rational function.

H

Graph of a rational function.

I

Graph of a rational function.

Give groups 2–3 minutes to take turns describing a graph and guessing. If partners finish early, ask them to play another round.

Here are some possible questions for discussion:

  • “What types of descriptions were helpful for identifying a graph?” (The location of the asymptote.)
  • “Graphs like A and B have the same asymptote. How did you know which one your partner was talking about?” (Graph A has values that get larger and larger in the negative direction as the \(x\) values get closer to the asymptote on the left, while Graph B has values that get larger and larger in the positive direction as the \(x\) values get closer to the asymptote on the left.)
  • “These graphs are either all positive or all negative depending on what side of the vertical asymptote they are on. Do you think that is true of all rational functions?” (No, because if we graphed something like \(y=10+\frac{10}{x}\), the line crosses the \(x\)-axis when \(x\) is -1.)

17.4: Cool-down - Homecoming T-shirts (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The distance \(d\) that an object moving at constant speed travels is based on the length of time \(t\) the object travels and the speed \(r\) of the object. Often, this relationship is written as \(d=r \boldcdot t\). We could also write the relationship as \(r=\frac{d}{t}\) or \(t=\frac{d}{r}\). Depending on what we want to know, one form of this relationship may be more useful than another.

For example, the distance across the English Channel from Dover in England to Calais in France is 33.3 km. The time in hours it takes for a boat to make this crossing can be modeled by the function \(T(r)=\frac{33.3}{r}\), where \(r\) is measured in kilometers per hour.

For very small values of \(r\), the journey takes a long time. For larger values of \(r\) (and a fast boat!), the trip is shorter. The graph of the function shows how the travel time decreases as the speed of the boat increases.

Graph of a decreasing polynomial function through 3 points on a coordinate plane, origin O.

Unlike the graphs of polynomial functions that look smooth and connected, the graphs of some rational functions can look like separate pieces. For example, here is a graph of \(f(x)=\frac{1}{x-3}\).

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

The dashed line at \(x=3\) is a representation of a vertical asymptote. As \(x\) gets closer and closer to 3, think about what happens to the value of the expression for \(f(x)\). If we divide 1 by a very small negative number, we get a very big negative number, which is what happens on the left side of the vertical asymptote. If we divide 1 by a very small positive number, we get a very big positive number, which is what happens on the right side of the vertical asymptote. It is important to note that the drawn-in asymptote is not actually part of the graph of the function. Instead, it is a helpful reminder that the function has no value at \(x=3\) and very large absolute values at inputs very close to \(x=3\).