What is a Horizontal Asymptote?
A horizontal asymptote is an imaginary horizontal line on a graph. It shows the general direction of where a function might be headed. Unlike vertical asymptotes, which can never be touched or crossed, a horizontal asymptote merely shows a general trend in a certain direction.
How to Find a Horizontal Asymptote of a Rational Function by Hand
In order to find a horizontal asymptote for a rational function you should be familiar with a few terms:
- A rational function is a fraction of two polynomials like 1/x or [(x – 6) / (x2 – 8x + 12)])
- The degree of the polynomial is the number “raised to”. For example, second degree (x2), third degree (x3) or 99th degree (x99).
- The coefficient is the number before the “x”. For example, the coefficient of 5x2 is 5; the coefficient of 102x3 is 102.
How you find the horizontal asymptote depends on what you function/equation looks like: compare the highest degree polynomial in the numerator with the highest degree polynomial in the denominator. Choose one of the following:
They are the same degree.
The denominator has the highest degree.
The denominator has the lowest degree.
Divide the coefficients of the terms with the highest degree.
The highest degree terms (i.e. the terms with the highest power) are 8x2 on the top and 2x2 on the bottom, so:
8/ 2 = 4.
There is a horizontal asymptote at y = 4. The following graph confirms the location of the asymptote:
If the polynomial in the denominator has a higher degree than the numerator, the x-axis (y = 0) is the horizontal asymptote. For example, the following graph shows that the x-axis is a horizontal asymptote for 8x2/2x4 :
If the polynomial in the denominator is a lower degree than the numerator, there is no horizontal asymptote.
How to Find Horizontal Asymptotes on the TI-89: Steps
Note: Make sure you are on the home screen. If you aren’t on the home screen, press the Home button.
Step 1: Look at the exponents in the denominator and numerator.
If the largest exponent of the numerator is larger than the largest exponent of the denominator, there is no asymptote. That’s it! You’re Done!
If the largest exponent of the denominator of the function is larger than the largest exponent of the numerator, go to Step 2.
If the exponential degrees are the same in the numerator and denominator, go to Step 3.
Step 2: The horizontal asymptote will be y = 0. That’s it! You’re done!
Step 3: Enter your function into the y=editor. For example, you might have the function f(x) = (2x2 – 4) / (x2 + 4). To enter the function into the y=editor, follow Steps 4 and 5.
Step 4: Press the diamond key and then F1 to enter into the y=editor.
Step 5: Enter the function. For example, if your function is f(x) = (2x2 – 4) / (x2 + 4) then press ( 2 x ^ 2 – 4 ) / ( x ^ 2 + 4 ) then ENTER.
Step 6: Press the diamond key and F5 to view a table of values for the function.
Step 7: Scroll far down the table and look the y values. You will notice that as x increases, the graph gets closer and closer and closer to y=2 but does not reach this value. The graph even hits y=1.999999. The horizontal asymptote is y=2.