Lesson 17

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.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.