This one isn't particularly revolutionary, but addresses a concern we had in the Pre-Calculus breakout session at TMC13: what's with rational functions? Myself, @calcdave, and @MaryBourassa discussed some ideas for addresses them and their relevance to Calculus. Mary showed off a neat matching activity she had, and Dave tried to spit ball some conceptual ideas.
In the end we had two conclusions: for Calculus purposes, the place of a rational function is to introduce the ideas of asymptotes and other discontinuities. Well, and holes are dirty dirty lies. But that's another story.
I wrote this down and made a note to include them in Pre-Cal this year (previously I skipped them). When the time came, I was still a bit stuck. Then I got hit with some dual inspiration. The preceding lesson was function compositions [f(x) + g(x)] and all that. A wide eyed student asked "well what about dividing?" And I was kinda like "well, yeah, that's a thing, but it's weird." Hours later was the DUH moment as I remembered this Nowak post I had read forever ago.
Rationals as Compositions
Inspiration in hand, I drew up a simple little desmos activity.
Part 1 was a simple little visual for f(x) + g(x) and f(x) - g(x). It's not that exciting. Here's the full write up.
The second part isn't bad though. A great desmos feature is that it does a little more to explain problem points with functions like this. A TI-84 will just flash a sorta but not really helpful ERROR in the table (and jack all on the graph). Students explored the three pairs (setting the table and rules themselves), noting anything "odd" in the third column. Would look something like this:
I had them share odd values confirming that they were getting the same weirdness for a given function pair.
Then the math part. We defined those weird points as one of two things: asymptotes or removable discontinuities. How can you tell the difference? We reviewed factoring quadratics and they could see the matching (x+3) in the numerator and denominator. This also the point where we defined rational functions as the ratio of two independent functions, simultaneously answering the "what about dividing?" question from before.
For assessment purposes, I gave them some ratios, asked for points of interest and had them make simple statements about them. "There is a ________ at x=____." Nothing crazy.
As one last step I discussed how to graph such things, paying attention to label asymptotes and holes. Despite all the naysaying, the TI-84s were of use here. I showed how to enter the ratio into Y=, locate the asymptote on the table, and graph points on either side to generate a graph. The work we did previous to this really helped. Now that they knew how to tell the difference between an asymptote and hole, they understood why the TI threw up multiple ERROR points.
I could've done more. For example, we didn't try to graph anything with multiple asymptotes (though they were given some ratios for practice that had them), or mess with anything other than quadratics. This being Pre-Calculus though, there was a clear goal: prepare them better for limits. Holes and asymptotes are all over the first units of Calculus, hopefully more will be familiar with the idea should they go on to Calculus.