Practice with Asymptotes

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SWBAT to identify discontinuities and asymptotes of rational functions. SWBAT simplify and solve rational expressions and equations.

Big Idea

Today, we'll aim for efficiency and accuracy as we practice solving problems concerning rational functions.

Reviewing our Progress

15 minutes

Yesterday was a day for making arguments and drawing conclusions; today will be a day for synthesizing and putting things into practice.  So, I'll begin with by asking the class to remind me what conclusions we came to yesterday.

As they explain the patterns that we had identified regarding vertical and horizontal asymptotes, I'll summarize them on the board.  The precise wording (MP 6) should be something that makes sense to the students (this is for their notes), but what I want to convey are the ways to find vertical asymptote and horizontal asymptotes.  Depending on the results of the exit ticket from the previous lesson, this discussion may take more or less than 15 minutes.

I'll also quickly review the process for identifying discontinuities, simplifying rational expressions, and solving rational equations. 

Practice with Asymptotes

30 minutes

Now, for practice I'll assign Practice with Asymptotes.

For this practice, I'll have the students working in small groups and I'll encourage working on the problems efficiently by correctly applying the rules we've discovered. (MP 7) I'll set the stage for this with a challenge for students to "First predict the behavior of this function with a few simple tests and justify it to each other.  Make a sketch of what you think the graph should look like, then confirm your conjectures with a graphing calculator."  This habit of first making predictions and then testing them with technology is very important for reasons I discuss in this video: Why graph by hand?.

If my students need an added challenge (and some practice solving rational equations), I may ask them to find the intersection point(s) of the functions in #2 and #3 and of the functions in #2 and #6.  Watch out for extraneous solutions!


2 minutes

During this lesson, I've had lots of time to observe my students progress and determine just how well everyone is doing.  I expect that most students will have only a few problems left, so tonight's homework will be to complete the assignment.  I'll also encourage everyone to check their solutions in the back of the book for all the odd problems.  If they find they've made a mistake, they need to go back and correct it.  Too many students don't take advantage of this opportunity to self-correct, and it's important for them to get in the habit of verifying their solutions (in whatever way they can).  When they find a mistake, they should persevere in the search for the correct solution! (MP 1)