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.
Values for f(x) and g(x) | Which polynomial has the higher degree? | Is there a horizontal asymptote? |
f(x) = x g(x) = x2 |
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f(x) = x2 − 9 g(x) = x |
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f(x) = x + 2 g(x) = x2 + 10x + 21 |
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f(x) = x(x + 2)(x − 3) g(x) = (x − 9)(x + 4) |
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. |
What is the horizontal asymptote when the degree of the numerator is greater?
What is the horizontal asymptote when the degree of the numerator is less than the denominator?
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.
As you changed the values of the coefficients for functions with a numerator and denominator of equal degree, where is the horizontal asymptote located?
Will it be possible to have a horizontal asymptote at y = 0 when the degrees of the numerator and denominator are the same?
As you worked through this section, you found that there were three scenarios that could describe the horizontal asymptote. They are as follows:
What do each of those scenarios mean for the horizontal asymptote of a rational function?
Will holes, x-intercepts, and vertical asymptotes have an impact on determining the horizontal asymptote?
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)
2. Where is the horizontal asymptote of f(x) = 3 over x + 1 3 x + 1 ?
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