Fresh eyes on end behavior of rational functions

Over the past few years of teaching precalculus regularly, I’ve experimented with a variety of approaches to rational functions in an attempt to find one that will result in students discovering and eventually truly understanding how the equation of a rational function determines its graph. I have never felt very successful. Too many students get confused somewhere along the way, come to feel like there are just way too many possibilities, particularly when it comes to end behavior, and—after arriving at the conclusion that it’s not worth the effort they perceive as being required to understand why differing degrees in numerator and denominator produce different end behavior—resort to trying to memorize three possibilities relating to horizontal asymptotes, i.e., same degree in numerator and denominator means take the ratio of leading coefficients to find the horizontal asymptote, denominator greater degree means the horizontal asymptote is $$y=0$$, and numerator greater degree means there’s no horizontal asymptote. (In their defense, they are encouraged in this approach by textbooks which list these three possibilities in a “pay-attention-this-is-what-you-really-need-to-know” box.)

When trying to convince people that they can understand what’s going on, my “why” explanation has typically relied on their knowledge of of the end behavior of power functions:  “If we look at the function $$f(x)=\dfrac{20x^2}{ 5x^2+1}$$ and imagine moving past the place where x = 100, past the place where x = 1000, and continue on and on to huge x-values, that 1 in the denominator keeps having less and less influence and the function looks more and more and more like $$f(x)=\dfrac{20x^2}{5x^2}$$, which is just the constant function 4 [as long as x isn’t zero, which it isn’t when it’s 10 trillion].” While students tend to be okay with this, the fact that they don’t really internalize the idea becomes apparent when they have so much more trouble with the extension, “And if you change that ² in the denominator to a ³, then, as x gets huger and huger the function $$g(x)=\dfrac{20x^2}{ 5x^3+1}$$ starts to look indistinguishable from $$\dfrac{4}{x}$$, so the end behavior of this rational function is the same as the end behavior of $$\dfrac{1}{x}$$ .”

This year, however, things turned out so much better that I thought it was worth my first post to this blog in almost two years. By the end of our study of rational functions this time around, it was clear—not just from the tests, but from the quality of discourse as well—that these students understood end behavior better than any group I’d had before. While my emphasis on using limit notation and thinking about end behavior of functions starting on the second day of school probably helped, it was a great insight from a student (inspired, at least in part, by a new first activity on rational functions) that was the big game changer.

It was over a month ago now, and I don’t remember all the details of our momentous first class on rational functions, but we were investigating their asymptotes using this GeoGebraTube construction.

We started with the default function in the construction, where the degree of the numerator and denominator are equal, and we began experimenting with changing coefficients to see how the graph was affected. Once students became confident that the ratio of the leading coefficients determined the location of the horizontal asymptote, I changed the degree of the denominator to be larger than that of the numerator, and suddenly the horizontal asymptote was no longer where they expected it to be. We did some more experimenting, changing degrees and coefficients as students liked, and they soon became confident that if the degree of the numerator was less than that of the denominator, then the horizontal asymptote would be $$y=0$$. When I asked why, a student gave an answer that I wasn’t expecting. He said, “It’s because if you make the highest exponent in the numerator the same as the highest one in the denominator you get $$\dfrac{0x^3+6x^2-x-2}{2x^3-7x+3}$$ and $$0$$ divided by $$2$$ is $$0$$.”

Almost as soon as people started thinking about this, someone asked something like, “So if the degree of the numerator is bigger, does that mean there’s no horizontal asymptote because anything divided by zero is undefined?” Great satisfaction when experimenting in GeoGebra suggested this was indeed the case. And we were off to the exploration of slant asymptotes. And, “Can a parabola be an asymptote?” This student’s way of looking at the algebra of the problem, elegantly connecting the various graphical possibilities to ultra-familiar arithmetic, provided a way for people to see all the different possibilities for horizontal asymptotes or lack thereof as a single concept. (And, though, I didn’t think we were quite ready to go there, calculus students may appreciate the extension to the $$\dfrac{0}{0}$$ indeterminate form that comes when you imagine making the highest exponent of the numerator and denominator the same but greater than the degree of either.)

I have no idea why I never thought of it this way before. To someone well-steeped in rational functions and limits, it’s completely obvious once it’s pointed out, and while not all of the other students saw it as quickly or as clearly as the one who initially articulated it, many understood it within a couple of minutes, and it was a way of thinking about end behavior that almost all of them eventually internalized.  There are probably many teachers out there who’ve been using this explanation for years, but if you happen to be one like me who hasn’t, I highly recommend it, and if you’re lucky, you’ll get a student to say it first.