Finding Asymptotes of Rational Polynomial Functions

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  • 0:04 Asymptotes & Rational…
  • 1:02 Finding Vertical Asymptotes
  • 1:49 Finding Horizontal Asymptotes
  • 4:20 Finding Oblique Asymptotes
  • 5:09 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will go over asymptotes, rational functions, and the different types of asymptotes of rational functions. Through definition, properties, and examples, we will look at how to find the asymptotes of a rational function.

Asymptotes & Rational Functions

Suppose you're going for a walk along a trail lined with poison ivy. There is a river running next to the trail that you are trying to video as you walk along the trail. You want to get as close to the edge of the trail as possible for the best view of the river, but you don't want to touch the edge because of the poison ivy. You just walk along, getting closer and closer to the edge without actually touching it. This scenario gives us an idea of how a graph approaches an asymptote.

An asymptote is a line that the graph of a function approaches but never actually touches. There are vertical asymptotes, horizontal asymptotes, and oblique asymptotes. Oblique asymptotes are also called slant asymptotes. Vertical and horizontal asymptotes are vertical and horizontal lines, respectively. An oblique, or slant, asymptote is an asymptote that is neither vertical nor horizontal.


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Asymptotes are a well-known characteristic of rational functions. A rational function is a function that is a quotient of functions.


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Let's look at how to find the asymptotes of a rational function.

Finding Vertical Asymptotes

As we said, there are three types of asymptotes. First, let's look at how to find the vertical asymptotes of a rational function. To find vertical asymptotes, we want to follow these steps.

  1. Set the denominator equal to zero and solve.
  2. The values that make the denominator zero are where you will have vertical asymptotes.

For example, consider the function h(x) = x / (x 2 - 4). To find the vertical asymptotes, we would set the denominator equal to zero and solve. As we can see, we have x 2 - 4 = 0 to start out, and then we set each factor to be equal to zero with (x + 2) multiplied by (x - 2) equaling zero. Then we solve for both equations.


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And thus, the vertical asymptotes of the function are x = -2 and x = 2.


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Finding Horizontal Asymptotes

Horizontal asymptotes can be identified in a rational function by examining the degree of both the numerator and the denominator. The degree of a function is the highest exponent of the function. There are three possibilities in a rational function:


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Once we figure out which of these is true for our function, we can find the horizontal asymptotes. It's important to note that a graph can cross a horizontal asymptote. When this is the case, the graph approaches the asymptote but doesn't touch it on one part of the graph, instead crossing it somewhere else.

Let's look at each of these cases we've just mentioned.

The first case happens when the degree of the numerator is less than the degree of the denominator. When this is the case, y = 0 is a horizontal asymptote. The example given is the function we looked at when we found our vertical asymptotes, h(x) = x / (x 2 - 4). In this function, the degree of the numerator is 1, and the degree of the denominator is 2. Since 1 < 2, y = 0 is a horizontal asymptote of this function.


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The second case applies when the degrees of the numerator and denominator are equal. When this is the case, we need to deal with lead coefficients. The lead coefficient of a function is the number in front of the term with the highest exponent. For example, in the function r(x) = 2x 4 + 4x 3 - x + 5, the lead coefficient is 2, the number in front of x 4.

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