My algebra 2 class started rubbing elbows with Calculus this past month. Do the students know how close they got to the infinite God? Do they see that their discovery of an asymptote of a rational function is the beginning of the study of calculus? No. Except that I told them. I cannot let an opportunity like that pass me by. If I have a chance to instill a bit of wonder about the God who not only gave us the problem of instantaneous change, but also gave us the solution, then I am going to take it.
Rational algebraic functions often have asymptotes in their graphs, and these happen at moments where as the denominator of a fraction approaches zero, the value of the function suddenly shoots off into infinity. It’s like we broke the function. But wait, it gets weirder. The input value on the other side of the limit actually continues, but not in the path toward infinity, rather in the negative version of the original. For example, if a rational function has a value of 1 at the input of 3, and at input 2 it shoots to infinity, then what do you think happens at 1? Our logical brains might think it is closer to infinity, but it isn’t. Rather it’s -1! It’s the negative of the value that we got when we had 3. Don’t believe me? Try it yourself with this function: f(x) = 1/ (x-2)
The function flips at this asymptote. As in physics when we move a lens from one side of its focal point to the other, the image flips upside down or vice versa. It’s like a magic trick! Magic that catches our breath and causes us to think that for just one second, we touched infinity, and were sent back to earth just as quick