How to Find Asymptotes of Graphs

This post is all about finding  Vertical and Horizontal asymptotes of graphs.

Vertical Asymptotes

Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, \displaystyle y=\frac{2}{x-3}. 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. x-3=0, so x=3.

\displaystyle y=\frac{2}{x-3}

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. y=\ln (x+2).

\ln 0 is undefined, so when x+2=0 or x=-2, there will be a vertical asymptote at x=-2.

y=\ln (x+2)


Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as x 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, \displaystyle y=\frac{1}{x}+1. 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 y=1.

\displaystyle y=\frac{1}{x}+1

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, y=e^{-x}+1. When x is very large, e^{-x} will be very small, and hence y approaches 1. This means that the Horizontal Asymptote will be y=1.


Note: The graphs above were drawn using the software Geogebra. 🙂

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra

Author: mathtuition88

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