At the end of yesterday’s ultramarathon lesson, students should have a good understanding about why the graph had an asymptote and why Lily will never get her average pace at or below 10 minutes. Yesterday was the first type of rational function, where the degree of numerator and denominator are the same. Today we are going to use different contexts to explore the other two types of rational functions.
Give groups of students the task worksheet and have them work on the two problems for about 15 minutes. The gummy bear task should be very straightforward; I think it will make a lot of sense why the asymptote for the function will be at y = 0 (Gummy Bear Graph). That’s the upside of using these contexts to explore a topic that students usually find confusing – they have a conceptual understanding that can be transferred to a more efficient procedural understanding.
The homecoming example is a little more complex. Setting up the equation might be simple, but the graph will not be as intuitive to students because the slant asymptote. Hint: encourage students to use Desmos to graph this function instead of their graphing calculator; the end behavior and the asymptotes are much easier to see.
Once students have had enough time to grapple with the problems, pick two students to share their graphs with the class. Talk about the contexts of the problems and why there are asymptotes. Define rational functions and get them to see that all of these functions can be written as fractions. Use the context to think about why the asymptotes are a necessary part of the graph.
Next, think about the algebra of each function's equation. Ask students what happens as you plug in larger and larger values for x. Here are the targets that I want my students to get to:
1. For the ultramarathon equation, guide them to see that the 13 in the numerator affects the y value less and less as x gets larger. For large values of x, the y value will be pretty close to 10.
2. For the gummy bear problem, the denominator is getting huge while the numerator stays at five. The overall value of y gets closer and closer to zero.
3. For the homecoming task, the last term of 500/x will basically become zero as x gets really large. So the first two terms are the only factors that affect ticket price. Note: make sure you show students the original graph AND the graph of the asymptote on the same coordinate plane. For something that can be confusing, this visual is extremely powerful!
Now we want to use our three contexts to classify the three types of rational functions based on the degree of their exponents. It will be helpful to rewrite the homecoming equation as a single fraction. Students may not gravitate towards the degrees of the numerator and denominator determining what type of asymptote there is, so it is okay to guide them there.
We have been focusing on the horizontal and slant asymptotes, so now would be a good time to think about why the vertical asymptotes are where they are. My students usually have a very good grasp of how to find vertical asymptotes, but you may need to have that conversation.
The Rational Function Summary Sheet can be used to formalize much of the work that we have done over the past two days. Here is a video that explains what I envision will go in the table to summarize our work with rational functions:
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Finally, students can cement their knowledge by classifying the rational functions below the table and deciding what type of asymptote they should have. Since we have a context for each type of rational function, we can refer to them as ultramarathons, gummy bears, or homecomings. This will make the classifications into the three types a little more tangible for your students.