Horizontal Asymptotes & End Behavior

It's been a while since we've updated this blog, but we're back!

This is a lesson that we did way back in October (better late than never!) in my honors precalculus class. In this short activity students investigate what happens to rational functions toward infinity and negative infinity. The activity is fairly simple, but it helps kids improve number sense a whole lot more than memorizing "if the degree of the numerator=the degree of the denominator"....

Horizontal Asymptote Exploration

 


The "Quick Solve" was fairly straightforward for the students. It was easy for them to see that as x gets extremely large the "plus 4x" starts to matter less and less. As x approaches infinity the "minus 3" barely even matters at all. Once they move on to the rational functions they can start thinking about how the numerator and denominator grow in relation to one another, with just looking at the terms that has the most impact on the fraction.

I liked that this activity got the students away from trying to memorize rules to find horizontal asymptotes of rational functions. Once the students completed the table on page 1 we discussed as a class what these functions look like in the coordinate plane as you approach positive and negative infinity, ignoring what's happening between the extremes. This got us into the whole discussion of end behavior (they have already been working with limit notation) and the kids started to think about the "ends" of these functions when there was no horizontal asymptote.

All in all, I think this activity did its job - there's no real practical applications embedded here, but it makes the students just think about numbers in general; something I've been really trying to push with my students this year.