Horizontal Asymptotes and Holes: some comments about rational function graphs
When tutoring math 12 or calculus, you encounter graphs of rational functions. Let’s look at a couple of features:
Rational function graphs are defined by (and you get marks for) the locations of the asymptotes (if any), as well as the x and y intercepts (once again, provided they exist) and holes (if any). Today, we’ll look at two of these features: horizontal asymptotes and holes.
First, to holes: consider the following rational function:
f(x)=((x-1)(x+2))/((x+2)(x-3)) eqn 1
You can see that, since (x+2) is both in the top and the bottom, the function simplifies to
f(x)=(x-1)/(x-3) eqn 2
However, when you cancel, you are really dividing. Since you can’t divide by zero, you can’t cancel x+2 when x=-2. At x=-2, the equation remains undefined. Therefore, you will get a hole there. The graph of eqn 1, above, will follow the graph of eqn 2 identically, except for a hole at x=-2.
Now, to horizontal asymptotes: you get them when the degree on top matches the degree on the bottom or is less than the degree on the bottom.
Case 1: the degrees on top and bottom are the same. Consider
f(x)=(2x^2 – 2x -3)/(x^2 +17x+11)
To get the horizontal asymptote, divide the coefficient of the highest exponent term on top by the coefficient of the highest exponent term on the bottom. The horizontal asymptote will be y=2/1, or just y=2.
Case 2: the degree on the bottom is greater than that on top.
Simple: the horizontal asymptote is y=0.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC