This post is all about finding Vertical and Horizontal asymptotes of graphs.
Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, . When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there.
Hence, to find the asymptote, let the denominator be 0. E.g. , so .
Another way vertical asymptotes can come about is via logarithmic graphs, e.g. .
is undefined, so when or , there will be a vertical asymptote at .
Horizontal asymptotes usually come about when one of the terms approaches zero as approaches infinity.
To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number).
For example, . When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is .
Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, . When x is very large, will be very small, and hence approaches 1. This means that the Horizontal Asymptote will be .
Note: The graphs above were drawn using the software Geogebra. 🙂
Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra