Finding Vertical Asymptotes
Description/Explanation/Highlights
Video Description
In this video we will learn how to locate vertical asymptotes in a rational function.
Steps and Key Points to Remember
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.
Video Highlights
- 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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