Vertical Asymptote: Rules, Step by Step Examples

TI 89 Calculus > Vertical Asymptotes

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.
• A rational function is a fraction of two polynomials like 1/x or [(x – 6) / (x2 – 8x + 12)].)

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) = (x<sup2< sup=””>) / (x2 – 8x + 12)</sup2<>

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:

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!