Graphs of Rational Functions (Lesson 2.6)
Unit 2 Day 8
CED Topic(s): 1.8, 1.9, 1.10
Identify key features of a rational function including its domain, intercepts, holes, and vertical asymptotes from its graph and equation in factored form.
Use one-sided limit notation to describe the behavior of a rational function near a vertical asymptote.
Determine the y-value of a hole by examining function outputs at input values sufficiently close to the x-value of the hole.
Quick Lesson Plan
Today students look at rational functions from a more analytical perspective and think about how zeros, holes, and vertical asymptotes are related to one another and how they are represented in an equation and graph.
Allow students opportunities to make their own conjecture about when a hole versus a vertical asymptote will occur and avoid rushing in too early to summarize the findings. As students work on question 7, consider playing devil’s advocate and arguing against whoever the students agree with. Sound as convincing as possible! “I thought we always learned that if the numerator is zero then we have a zero of a function, there’s no reason it shouldn’t be a zero!” or “But we’re dividing by zero, is that even allowed??
Why does your calculator say ERROR at x=-1 and x=3?
Isn’t 0 over anything 0? Why is there not an x-intercept at x=3?
How would you find the x-intercepts if you didn’t have a graph?
How would you find the vertical asymptotes if you didn’t have a graph?
How could we figure out what value the function is getting closer and closer to at x=3 if we can’t actually evaluate the function at x=3?
This lesson, like yesterday’s, is foundational to students’ understanding of limits in AP Calculus. Students must understand the important difference between a function output of 0/k, k/0 and 0/0 for some constant k. The first denotes a zero of a function, the second a vertical asymptote, and the third an indeterminate form which requires more exploration.
On a technical note, students may be quick to cancel the identical factors in the numerator and denominator but emphasize to students that there is one important difference between the simplified version and the original version, namely what’s happening at x=3. Have students graph both on their calculator and compare the two graphs. They should look identical except for the hole at x=3. Note that the two functions are not exactly equivalent so it is technically incorrect to set them equal to each other. You can say that the “simplified” version has very similar behavior to the original function, EXCEPT at x=3. When finding the y-value of the hole, remember that the y-value is not the output of the function at the given x-value (since the function is undefined there) but simply the vertical location of the hole. In this course we encourage students to explore output values near the location of a hole, instead of evaluating the “simplified” function.
In the margin notes, add limit notation to describe the functions behavior to the left and right of a vertical asymptote and hole. Since one-sided limits are new to students, spend a couple minutes teaching students how to say the limit expression out loud correctly. The limit as x approaches -1 from the left of g(x) is negative infinity.