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Consider the rational function
One way we can think about these values is to rewrite the expression for
Written this way, it’s easier to see that as
A dashed line at
The line
More generally, if a rational function
Display the opening statement and graph for all to see. Give students 1 minute to read the Task Statement, and then ask, “In the expression for
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.
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Collect the language that 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. Display words and phrases, such as “gets close to 4,” “never reaches,” and “almost flat.”
During work time, select students with clear explanations for the final question to share during the Activity Synthesis.
Let
If students only look at the graph and think that the end behavior is close to $0, consider asking:
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.
Direct students’ attention to the reference created using Collect and Display. Ask students to share how they solved the first 3 questions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.
Select previously identified students to share what the end behavior of the function says about the context. Then, ask students how we could rewrite
Once students understand how the new form was written, ask, “Which term in
Conclude the discussion by making sure students understand that not all rational functions have horizontal asymptotes (if needed, display the graphs of the cylinder from an earlier lesson), but those that do can be written a certain way, which is the focus of the next activity.
Here are four graphs of rational functions.
AGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma 4), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
BGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma -4), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
CGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma 1), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
DGraph of a rational function f(x) with a dashed horizontal asymptote through (0 comma -1), on xy-plane. Each axis from -10 to 8, by 2’s. Vertical asymptote of the function at x = 0.
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 to 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.