Asymptotic Behavior, Day 1 of 2

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SWBAT describe the behavior of a rational function in terms of vertical and horizontal asymptotes.

Big Idea

Rational functions are asymptotic because strange things happen when you divide by very large or very small numbers.

What have we seen?

15 minutes

Class will begin with a summary discussion to evaluate our progress so far.  I'll simply have students share out their thoughts and describe things they've noticed, but I'll direct the conversation toward the three main points listed below.

"We have been studying rational equations and their graphs for some time now.  What have we seen?"

  1. End Behavior: Graphs of rational functions often approach a horizontal asymptote.  Numerically, this means that the value of the functions seems to get closer and closer to some fixed number as the value of x gets larger and larger.
  2. Discontinuity: Graphs of rational functions are often broken into separate branches.  The discontinuities are sometimes a single point (removable) and sometimes a vertical asymptote.
  3. Symmetry:  The branches of the graph of a rational function are often symmetrical in some way.

Today, we will focus on the two types of asymptotes: vertical & horizontal.  Specifically, our goal is to explain how these asymptotes can be identified simply by examining the equation. (MP 7)  This is a (+) standard in the Common Core, but I think it's an important one for my students.  See this video to hear why.

Making Predictions Based on Structure

30 minutes

Handout the worksheet Asymptotic Behavior.

The purpose of this problem set is to help students identify the ways that the structure of a rational function can help them predict its behavior.  Students will investigate a number of different functions and record either the presence or absence of asymptotes, as well as the equations for those asymptotes.  By studying the data, they should be able to come up with some general rules (MP 7).  Since students will not work at the same pace, I will have them begin individually, but move into groups of three after about 10 minutes.

First, they should identify points of discontinuity by factoring the denominator.

Second, they should determine whether the function has the form 0/0 or #/0 at those points.

Third, for any points at which the function is 0/0, they should simplify the function and re-evaluate.  Wherever the function is #/0 in its simplified form, there is a vertical asymptote.

Fourth, they should compare the highest-degree terms in the numerator and denominator.  These terms will not only indicate the presence of a horizontal asymptote, but also its value.

In order to make this as efficient as possible, all the students should have graphing calculators.  Alternatively, you could give each student a copy of Graphs for Asymptotic Behavior and give them the additional task of matching each function to its graph.

Wrapping Up

5 minutes

In most cases, the class is going to need more time to complete the investigation and draw out the associated patterns.  As class draws to an end, I'll make a note of how much progress most students have made.  For homework, I'll ask everyone to complete one more row of their table.  They should come to back to class tomorrow ready to show what they did on their own and compare their answers to those of their peers.