Calculus How To

Asymptote: Vertical, Horizontal & Oblique

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What is an Asymptote?

An asymptote is a line on a graph which a function approaches as it goes to infinity. The distance between the graph of the function and the asymptote approach zero as both tend to infinity, but they never merge.

asymptote

A vertical asymptote. The function approaches this line; Although it looks like it touches, it never actually does.

Types of Asymptote

There are three types of asymptotes:

  1. A horizontal asymptote is simply a straight horizontal line on the graph. It can be expressed by y = a, where a is some constant. As x goes to (negative or positive) infinity, the value of the function approaches a.

    A horizontal asymptote.

  2. A vertical asymptote is a vertical line on the graph; a line that can be expressed by x = a, where a is some constant. As x approaches this value, the function goes to infinity.
  3. An oblique or slant asymptote is, as its name suggests, a slanted line on the graph. More technically, it’s defined as any asymptote that isn’t parallel with either the horizontal or vertical axis. It can be expressed by the equation y = bx + a. As x approaches infinity, the graph of the function approaches this line.

    An oblique asymptote.

A function can have any number of vertical asymptotes: even an infinite number. It can only have two horizontal asymptotes.

Functions don’t cross their vertical asymptotes, but they may cross their horizontal asymptotes.

1. Horizontal Asymptotes

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 just 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:

  1. They are the same degree.
  2. The denominator has the highest degree.
  3. The denominator has the lowest degree.

1. Polynomials are the same degree

Divide the coefficients of the terms with the highest degree.
Example:



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:


2. The denominator has the highest degree.

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 :


Graph of (8x2)/(2x4) with the horizontal asymptote highlighted in yellow.


3. The denominator has the lowest degree.

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.

2. Vertical Asymptote

A vertical asymptote is a vertical line on a graph of a rational function.

  • An asymptote is a line that a function approaches; Even though it might look like it gets there on a graph, it never actually reaches that line. Asymptotes can be vertical (straight up) or horizontal (straight across).
  • A rational function is a fraction of two polynomial functions like 1/x or [(x – 6) / (x2 – 8x + 12)].)

vertical asymptote

How to Find Vertical Asymptotes

In any fraction, you aren’t allowed to divide by zero. This includes rational functions, so if you have any area on the graph where your denominator is zero, you’ll have a vertical asymptote.

To find out if a rational function has any vertical asymptotes, set the denominator equal to zero, then solve for x.

Example by Hand

Find where the vertical asymptotes are on the following function:
f(x) = (x2) / (x2 – 8x + 12)

If you set the denominator (x2 – 8x + 12) equal to zero, you’ll find the places on the graph where x can’t exist:

  • Factoring (x2 – 8x + 12) =
  • (x – 2)(x – 6)
  • x = 2 or x = 6

(If factoring isn’t your strong point, brush up with 30 minutes free tutoring with Chegg ).

Graphing the function (I used the free HRW graphing calculator), we can see that there are, as expected, vertical asymptotes at x = 2 and x = 6:
asymptotes


If you can’t solve for zero, then there are no vertical asymptotes. For example, let’s say your denominator is x2 + 9:
x2 + 9 = 0
x2 = –9
cannot be solved.

Vertical Asymptote Steps on the TI89

If you have a graphing calculator you can find vertical asymptotes in seconds.

Example problem: Find the vertical asymptote on the TI89 for the following equation:
f(x) = (x2) / (x2 – 8x + 12)

Note: Make sure you are on the home screen. If you aren’t on the home screen, press the Home button.

Step 1: F2 and then press 4 to select the “zeros” command.

Step 2: Press (x^2)/(x^2-8x+12),x to enter the function.

Step 3: Press ) to close the right parenthesis.

Step 4: Press Enter.

Step 5: Look at the results. The resulting zeros for this rational function will appear as a notation like: (2,6) This means that there is either a vertical asymptote or a hole at x = 2 and x = 6.

Step 5: Plug the values from Step 5 into the calculator to mark the difference between a vertical asymptote and a hole. The numerator is x-6, so press 2, -, -4 and then press Enter to get 6. This means that f(2) = 6, confirming there is a vertical asymptote at x = -4. When x = 0, the numerator is equal to -6. This confirms that there is a hole in the graph at x = -6. If the numerator is ever equal to zero, this means that there is a hole in the graph and not a vertical asymptote.

That’s How to Find a Vertical Asymptote on the TI89!

You can double check your answer with this calculator by Symbolab.

3. Oblique Asymptotes

oblique asymptoteAn oblique asymptote (also called a nonlinear or slant asymptote) is an asymptote not parallel to the y-axis or x-axis.



You have a couple of options for finding oblique asymptotes:

  1. By hand (long division)
  2. TI-89 Propfrac command

1. By Hand

You can find oblique asymptotes by long division. This isn’t recommended, mostly because you’ll open yourself up to arithmetic and algebraic errors by hand. But, if you are required to find an oblique asymptote by hand, you can find the complete procedure in this pdf.

2. TI-89

You can also find nonlinear asymptotes on the TI-89 graphing calculator by using the propFrac( command, which rewrites a rational function as a polynomial function plus a proper fraction. The parts of the proper fraction give you information about the nonlinear asymptotes for the function.

Example 1

Example Problem: Find the oblique asymptote for the following function:
f(x) = (x2 – 3x + 5) / (x + 4)

Step 1:
Press the HOME key

Step 2: Press F2 and then 7 to select the “propFrac(” command.

Step 3: Press ( x ^ 2 – 3 x + 5 ) ÷ ( x + 4 ) ).

Step 4: Press the ENTER key.

The result is the sum of a proper fraction (33 / x + 4) and a linear polynomial function (x – 7). The linear function y = x – 7 is the equation of the oblique asymptote. You can use this method to find any oblique asymptote on the TI-89.

That’s it! You’re done!


Find Nonlinear Asymptotes: Example 2

Example problem: Find the nonlinear asymptotes for the function: f(x) = (x3 – 8x2 + x + 10)(x – 6)

Step 1: Press the HOME key.

Step 2: Press F2 and then press 7 to select the propFrac( command.

Step 3: Type the function into the calculator. To enter the function, press the following keys:( x ^ 3 – 8 x ^ 2 + x + 1 0 ) ÷ ( x – 6 ) ).

Step 4: Press the ENTER key.
The result is the sum of a proper fraction (-56x2 – 2x – 11) and a quadratic function (x2 – 2x – 11). The quadratic function y = x2 – 2x – 11 is the equation of the nonlinear asymptote. You can use this method to find any nonlinear asymptote on the TI-89.

That’s it! You’re done!

Tip: Makes sure you enclose the whole equation by parentheses, otherwise you won’t get the right result for the propfrac(command.

A Proper Fraction has a numerator that is smaller than its denominator and represents a quantity less than the whole, or < 1: 1/5, 2/5, 3/5, and 4/5 are proper fractions.

References

Asymptotes of Rational Functions. Retrieved September 16, 2019 from: https://www.austincc.edu/pintutor/pin_mh/_source/Handouts/Asymptotes/Horizontal_and_Slant_Asymptotes_of_Rational_Functions.pdf
Kmiecik, Joan. Get Your Asymptote Intercepts Here. The Mathematics Teacher, Vol. 83, No. 5 (MAY 1990), pp. 402-404 Published by: National Council of Teachers of Mathematics
Retrieved from https://www.jstor.org/stable/27966722 on September 21, 2018.
Osikiewicz, Beth-Allyn. Math 11012 Class Handout: Vertical and Horizontal Asymptotes. Retrieved from http://www.personal.kent.edu/~bosikiew/Math11012/vertical-horizontal.pdf on September 21, 2018.
Sterling, Mary Jane. Oblique Asymptotes. Supplement to Algebra II Workbook for Dummies. Retrieved from https://www.dummies.com/education/math/algebra/oblique-asymptotes/ on September 21, 2018

CITE THIS AS:
Stephanie Glen. "Asymptote: Vertical, Horizontal & Oblique" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/calculus-definitions/asymptote-vertical-horizontal-oblique/
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