Horizontal asymptotes of rational functions will help you describe the end behavior of a graph. Since this is true, unlike vertical asymptotes, it is possible for the graph of a rational function to cross a horizontal asymptote.

The location of the horizontal asymptote will depend on the degree of the polynomials that make up the numerator and denominator of the function. There are three possibilities when you compare the degrees of the numerator and denominator: one with either greater than, less than, or equal to the other.

In the following applet, you will be able to determine how the horizontal asymptote is affected when the degree of the numerator is greater than the degree of the denominator and when the degree of the numerator is less than the degree of the denominator.

Click on the image to access the applet. Once you are in the applet, check the box "Show asymptote as x → ∞." This will help you see the horizontal asymptote, if it exists. Use the applet to complete the table below, and then use the table to answer the conclusion questions that follow.

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Values for f(x) and g(x)	Which polynomial has the higher degree?	Is there a horizontal asymptote?
f(x) = x g(x) = x2	g(x), which is the denominator	There is a horizontal asymptote at y = 0.
f(x) = x2 − 9 g(x) = x	f(x), which is the numerator	There is no horizontal asymptote.
f(x) = x + 2 g(x) = x2 + 10x + 21	g(x), which is the denominator	There is a horizontal asymptote at y = 0.
f(x) = x(x + 2)(x − 3) g(x) = (x − 9)(x + 4)	f(x), which is the numerator	There is no horizontal asymptote.

Conclusion Questions

What is the horizontal asymptote when the degree of the numerator is greater?

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There is not a horizontal asymptote.

What is the horizontal asymptote when the degree of the numerator is less than the denominator?

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For this set of functions, the horizontal asymptote is the x-axis, or the line y = 0.

The most common horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator. In all of the examples you worked in the previous section of the lesson, the horizontal asymptote was at y = 1. The following applet will help you determine how this relationship impacts the horizontal asymptote.

Click on the image to access the applet. Move the sliders for the values of a and b. These values represent the coefficients in the rational function y = ax + 1 over bx ax + 1 bx . The orange slider will move the orange line, which will help you locate the horizontal asymptote. Determine how the leading coefficients of the rational function impact the horizontal asymptote.

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Conclusion Questions

As you changed the values of the coefficients for functions with a numerator and denominator of equal degree, where is the horizontal asymptote located?

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When rational functions have the equal degree in the numerator and denominator, the horizontal asymptote can be found at y = leading coefficient of the numerator over lead coefficient of the denominator leading coefficient of the numerator leading coefficient of the denominator .

Will it be possible to have a horizontal asymptote at y = 0 when the degrees of the numerator and denominator are the same?

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This is not possible, since the horizontal asymptote in this case will be found at y = leading coefficient of the numerator over leading coefficient of the denominator leading coefficient of the numerator leading coefficient of the denominator . This would mean that the leading coefficients would have to be 0, which cannot be true.

Pause and Reflect

As you worked through this section, you found that there were three scenarios that could describe the horizontal asymptote. They are as follows:

• The degree of the numerator is less than the degree of the denominator.
• The degree of the numerator is greater than the degree of the denominator.
• The degree of the numerator is equal to the degree of the denominator.

What do each of those scenarios mean for the horizontal asymptote of a rational function?

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• The degree of the numerator is less than the degree of the denominator means that the horizontal asymptote will always be at y = 0.
• The degree of the numerator is greater than the degree of the denominator means that there is no horizontal asymptote.
• The degree of the numerator is equal to the degree of the denominator means that the horizontal asymptote is at y = leading coefficient of the numerator over lead coefficient of the denominator leading coefficient of the numerator leading coefficient of the denominator .

Will holes, x-intercepts, and vertical asymptotes have an impact on determining the horizontal asymptote?

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No. The horizontal asymptote will only be determined by the relationship between the degrees of the polynomials of the numerator and denominator.

Practice

1. Where is the horizontal asymptote of the following rational function?

h(x) = (2x - 1)(3x + 2) over (x + 1)(x - 2) (2x − 1)(3x + 2) (x + 1)(x − 2)

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Since the degrees are the same, look at the leading coefficients of the polynomials. You may need to multiply the factors to get the leading coefficients.

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The horizontal asymptote is at y = 6.

2. Where is the horizontal asymptote of f(x) = 3 over x + 1 3 x + 1 ?

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Look at the degree of the numerator and denominator, and see what that means for the horizontal asymptote.

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The horizontal asymptote is at y = 0.

3. Which of the following rational functions do not have horizontal asymptotes? There may be more than one answer.

I. f(x) = 2x squared + 2 over x + 1 2x² + 2 x + 1
II. f(x) = 3x over 9x + 1 3x 9x + 1
III. f(x) = x cubed + 2x squared + 5x + 1 x³ + 2x² + 5x + 1 x² + 4x + 4
IV. f(x) = x squared - 25 over x squared - 25 x² − 25 x² − 25

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Check the relationship between the degree of the numerator and the degree of the denominator.

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The rational functions that do not have horizontal asymptotes are I and III because the degree of the numerator is greater than the degree of the denominator.