# How to Find Vertical Asymptotes

An asymptote of a polynomial is any straight line that approaches the graph but never reaches it. Asymptotes can be slant (with a slope), horizontal, or vertical. A vertical asymptote is a vertical line that approaches but does not touch the graph of a polynomial. To find a vertical asymptote, scroll down to Step 1.

## Steps

- 1
**Make sure your polynomial is in p/q form.**The polynomial must have both a numerator and a denominator that contain at least one term of x.- For example, take the polynomial x^2 + 2x + 2 / x^2 + 2x – 8. This polynomial is in p/q form with at least one term of x, so you can look for a vertical asymptote.

- 2
**Find the roots of the denominator.**The roots of a polynomial are those values of x that, when substituted into your equation, result in 0. For a second order equation (an equation with no exponent higher than the power of 2), the roots can be determined with the quadratic formula:- In the example above, the denominator is x2 + 2x – 8. Put your values (a = 1, b = 2, c = -8) into the quadratic formula to find two roots (in this case, x = -4 and x = 2).

- 3
**Write down the asymptote equations.**When you find the roots of your denominator, your answers can be written as the equations of straight lines: x = root1, x = root2, and so on. These lines are actually vertical asymptotes of your polynomial.- In the example above, write your asymptote equations as x = -4 and x = 2.

- 4
**Draw the lines.**Draw the lines of your equations alongside the graph of your polynomial to verify that the lines are actually vertical asymptotes – that they approach the graph but do not touch it.- In the example above, the lines x = -4 and x = 2 approximate the graph of the polynomial, but, as shown below, they do not cross it. Therefore, x = -4 and x = 2 are both true vertical asymptotes of the polynomial.

## Tips

- The length of the x-axis should be kept small, so you can clearly see that the asymptote never touches the graph of the polynomial.
- In engineering, asymptotes are tremendously helpful, as they allow you to approximate linear behaviors, which are easy to analyze, for non-linear behaviors.

## Article Info

Categories: Algebra