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How to Find Horizontal Asymptotes

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  • 0:04 Asymptotes
  • 1:48 Finding Horizontal Asymptotes
  • 3:46 Examples
  • 5:19 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn rules and definitions that will enable us to find horizontal asymptotes. Through applying these rules and through examples, we will deepen our understanding of finding horizontal asymptotes.

Asymptotes

The following graph represents the annual cost of a car based on how many years it's owned. This annual cost can be represented by this formula:

C = (20,000 + 2,000n) / n

car example graph

Notice as the number of years the car is owned increases, it appears that the annual cost of the car approaches $2,000. The horizontal line C = 2,000 is what we call an asymptote, and it tells us that the longer we own the car, the closer our annual cost will get to $2,000.

In mathematics, an asymptote is a line that a graph approaches but never actually touches. Asymptotes show up in graphs of equations modeling population growth and decline, medicine, revenue and cost, as well as many other real world applications.

In this graph, the red lines are asymptotes.

graph with asymptotes

Here we see a vertical asymptote and a horizontal asymptote. Our concentration is going to be on horizontal asymptotes and how to find them.

Let's list the steps to finding horizontal asymptotes, and then we'll illustrate those steps through multiple examples. Let's quickly define two terms, so we can understand all the vocabulary in the steps and rules.

Degree of a polynomial: the highest exponent in that expression. For example, x^2 + x - 4 has degree two. Similarly, x^5 - x^7 + x^2 + 1 has degree seven. The degree of a constant number with no variables is zero.

Lead Coefficient: the number in front of the variable with the highest exponent. For example, 3x^4 +9x^2 - 5 has lead coefficient of 3. In another example, x + 5 has lead coefficient of one.

Finding Horizontal Asymptotes

Okay, let's follow the steps to finding a horizontal asymptote.

  1. Use algebra to isolate y on one side of the equation. Now you should have an equation where y is equal to an expression with a numerator and a denominator. For example y = (x + 3) / (2x - 4), or y = 3x^2 + 15. Although it doesn't look like there is a denominator in the expression 3x^2 + 15, the denominator is one. When we divide any expression or number by one, we get the same expression or number back out.
  2. Identify the degrees of the numerator and denominator, and determine if the degree of the numerator is greater than, less than, or equal to the degree of the denominator.
  3. Use the following rules to compare those degrees and find our horizontal asymptotes.

Rule 1

If the degree of the numerator is less than the degree of the denominator, then there is a horizontal asymptote at y = 0 (the x-axis).

Rule 2

If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

Rule 3

If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at y = (lead coefficient of the numerator)/(lead coefficient of the denominator).

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