topics rational functions 12 - My Math Education

Finding Vertical Asymptotes

In this video we will learn how to locate vertical asymptotes in a rational function.

To find a vertical asymptote in a rational function, follow these steps:

Vertical asymptotes occur when a graph approaches a vertical line but never reaches it.
To find a vertical asymptote, set the denominator equal to 0 and solve the resulting equation for x. The result (or results) is (are) the vertical asymptote(s).
For example, to find the vertical asymptote in the rational equation $y=\frac{\textstyle x+4}{\textstyle 2x-1}$ , set the denominator $2x-1=0$ . Adding 1 to both sides and dividing by 2 results in the equation $x=\frac{\textstyle 1}{\textstyle 2}$ so $x=\frac{\textstyle 1}{\textstyle 2}$ is the vertical asymptote.
In the example $y=\frac{\textstyle 3x+7}{\textstyle 9x^2-4}$ , the denominator is a quadratic because of the $x^2$ . This will result in two asymptotes.
Set $9x^2-4=0$ to find the asymptotes. To solve the resulting equation, factor the left side and set each factor = 0.
The factored version is $(3x-2)(3x+2)=0$ so $3x-2=0$ & $3x+2=0$ . Solving for x results in vertical asymptotes of $x=\frac{\textstyle 2}{\textstyle 3}$ and $x=-\frac{\textstyle 2}{\textstyle 3}$

Here are some key points to keep in mind when finding vertical asymptotes for rational equations.

Vertical asymptotes occur where denominators are undefined or equal to zero.
Vertical asymptotes are the solution that graphs a vertical line where the denominator equals 0.
Vertical asymptotes are often graphed as dotted reference lines on the graph.

00:00 Introduction
00:43 $y=\frac{\textstyle x+4}{\textstyle 2x-1}$ example of finding a vertical asymptote
02:15 $y=\frac{\textstyle 2x-6}{\textstyle 3x+9}$ second example of finding a vertical asymptote
03:02 $y=\frac{\textstyle 3x+7}{\textstyle 9x^2-4}$ example of finding a vertical asymptote when the denominator is a quadratic and there are two vertical asymptotes
05:25 $y=\frac{\textstyle 4x}{\textstyle x^2-5x+6}$ second example of finding a vertical asymptote when the denominator is a quadratic and there are two vertical asymptotes
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