Graphs of rational functions

I am a nerd, and if anything, this post will only go to support that fact, and show people how much of a nerd I really am. In my math class, we were talking about common functions and their graphs. One of them is a rational function, which has some sort of polynomial, divided by some sort of polynomial. The thing that makes a rational function fun (for me) is the fact that it has asymptotes, or certain parts of the graphs that it will never touch, but approaches. Take for example, if you had a rational function with x² - 1 as the denominator, there would be vertical asymptotes at x = 1, and x = -1. So as the function approaches 1, or -1, the graph of the rational function would approach the line x = 1, and x = -1, but never touch or cross it. There are also horizontal asymptotes and oblique asymptotes, depending on what degree the numerator is in relation to the denominator. Sometimes, and I'm not exactly sure of the rules on this, the function can cross the oblique or horizontal asymptote. You simply put the asymptote equal to the function, and solve for x, and if you come up with a number, it crosses that asymptote at that value of x.

Up until a few days ago, I thought that was it. But that's not all. When I was looking for material to study for my math test online (since I had forgotten my book), I found there are other asymptotes that are not linear. They exist for functions where the denominator is 2 or more degrees less then the numerator. This is found with long division of the rational function (maybe more on this later). But a function will approach these asymptotes that are non-linear as well, and sometimes cross them. Here is an example of a function with a non-linear asymptote that I have graphed using graphing software on the computer.

The non-linear asymptote in this case is the function y = x² + 4x + 9. And when you set this function equal to the rational function, the solution is x = -3/4, and you can see that the function crosses this asymptote at x = -3/4. (if you want a closer look at the graph, click on the picture, and it should open up a full size of it on another page). Anyways, I just found this interesting, that asymptotes didn't necessarily have to be linear.