Unit 3, Lesson 2
Graphs of Rational Functions (Part 1)
Algebra 2
2.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
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, and follow with a whole-class discussion.
Activity
NoneKiran’s aunt plans to bike 10 miles.
- How long will it take if she bikes at an average rate of 8 miles per hour?
- How long will it take if she bikes at an average rate of miles per hour?
- Kiran wants to join his aunt, but he only has 45 minutes. What will their average rate need to be for him to finish on time?
- What will their average rate need to be if they have hours?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to share how they reasoned about the questions in general and their thinking about the relationships between distance, rate, and time. Select 1–2 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 ”“ are both acceptable.
Lesson Synthesis
The purpose of this discussion is to give students an opportunity to practice describing the graphs of rational functions and 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 students 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 not use the name of the graph as part of their description. Display the following graphs for all to see.
A
B
C
D
E
F
G
H
I
Give groups 1-2 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 a brief follow-up 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 -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 -values get closer to the asymptote on the left.)
Student Lesson Summary
The distance that an object moving at constant speed travels is based on the length of time the object travels and the speed of the object. Often, this relationship is written as . We could also write the relationship as or . Depending on what we want to know, one form of this relationship may be more useful than another.
For example, the distance of a route crossing the English Channel from Dover in England to Calais in France is 33.3 kilometers. The time in hours it takes for a boat to make this crossing can be modeled by the function , where is measured in kilometers per hour.
For very small values of , the journey takes a long time since the boat is going very slow. For larger values of (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. When graphing, we also consider only positive values for and since both speed and time are positive in this situation.
Graph of a decreasing polynomial function, rt-plane, origin O. Horizontal axis labeled r, scale 0 to 40 by 5’s. Vertical axis labeled t, scale 0 to 40, by 2 point 5’s. Function decreases through (0 point 9 comma 37), through (2 comma 16 point 65) and through (36 comma 0 point 925). The function has horizontal and vertical axis as asymptotes.
Let’s consider the graph from the other direction—that is, think about the output of the function from right to left. As the input approaches , the output increases rapidly. This makes sense because the slower the boat goes, the longer the trip is going to take. But what about at ?
It turns out that even if we extended the graphing window vertically, we would never find an output for . A boat with a speed of 0 kilometers per hour will not cross the English Channel, so there is no output for the function at this input. This is an example of a feature of rational functions: a vertical asymptote.
Graph of a rational function f(x) with a dashed vertical asymptote through (3 comma 0), on xy-plane, origin O. Each axis from -10 to 8, by 2’s. Horizontal asymptote of the function at y = 3.
A vertical asymptote is a value at which the function is undefined and the outputs of the function as it approaches the value of the asymptote get larger and larger in either the negative or positive direction. Vertical asymptotes can give some rational functions a disconnected look. For example, here is a graph of .
The dashed line at is a representation of a vertical asymptote. As gets closer and closer to 3, think about what happens to the value of the expression for . 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 and very large absolute values at inputs very close to .