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 + 2,000*n*) / *n*

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.

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, 3*x*^4 +9*x*^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.

- 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) / (2*x*– 4), or*y*= 3*x*^2 + 15. Although it doesn’t look like there is a denominator in the expression 3*x*^2 + 15, the denominator is one. When we divide any expression or number by one, we get the same expression or number back out. - 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.
- 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).

If we look back at our car example, we stated that it looked like the asymptote was *C* = 2,000. Using what we’ve just seen, we can verify this. Our equation is *C* = (20,000 + 2,000*n*) / *n*. We see that the numerator and denominator both have degree one, making them equal, so Rule 3 applies. The lead coefficient of the numerator is 2,000 and the lead coefficient of the denominator is one, so we have a horizontal asymptote at *C* = 2,000 / 1, which is the same as *C* = 2,000.

Let’s continue with some more examples to get more comfortable with the material in this lesson.

## Examples

Determine if the graphs of the following equations have horizontal asymptotes. If so, then identify the horizontal asymptote.

a.) (*x* + 5)*y* = 5 – *x*

b.) (*x*^2 – 5*x* + 6)*y* = -2*x*

c.) 4*x* + (*x* + 2)*y* = –*x*^2

Solutions:

We’ll first do step 1 for all three problems. To isolate *y* in (a), we divide both sides by *x* + 5. To isolate *y* in (b), we divide both sides by *x*^2 – 5*x* + 6. Lastly, to isolate *y* in (c), we subtract 4*x* from both sides, then divide both sides by *x* + 2. This gives us the following:

a.) We see that the numerator and denominator both have degree one. Therefore the degrees are equal, so Rule 3 applies. The lead coefficient of the numerator is -1, and the lead coefficient of the denominator is 1. Therefore, the horizontal asymptote is *y* = -1/1, or *y* = -1.

b.) The degree of the numerator is one, and the degree of the denominator is two. Therefore the degree of the numerator is less than the degree of the denominator, so Rule 1 applies. Thus, there is a horizontal asymptote at *y* = 0.

c.) We see that the degree of the numerator is two, and the degree of the denominator is one. Therefore, the degree of the numerator is greater than the degree of the denominator, so Rule 2 applies, and there is no horizontal asymptote.

## Lesson Summary

An **asymptote** is a line that a graph approaches but never actually touches. Finding horizontal asymptotes is a matter of isolating *y*, comparing degrees, and applying the rules.

- 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).