Vertical Asymptote Study Guide

A vertical asymptote is a vertical line that a graph approaches but does not touch. It represents values that the function cannot take, leading to a discontinuity in the graph.

Finding Vertical Asymptotes

To find the vertical asymptotes of a function, look for values of x that make the denominator of a rational function (a fraction) equal to zero.
Set the denominator equal to zero and solve for x. The values of x obtained are the vertical asymptotes of the function.

Example

Find the vertical asymptotes of the function f(x) = (2x + 3) / (x^2 - 4).

To find the vertical asymptotes, set the denominator equal to zero:
x^2 - 4 = 0
(x - 2)(x + 2) = 0
x - 2 = 0 or x + 2 = 0
x = 2 or x = -2

So, the vertical asymptotes of the function are x = 2 and x = -2.

Graphing Vertical Asymptotes

When graphing a function with vertical asymptotes, draw dashed lines at the x-values of the vertical asymptotes.
Indicate that the graph approaches these lines but does not cross them, showing the discontinuity in the function.

Practice Problems

Find the vertical asymptotes of the function g(x) = (x^2 - 9) / (x - 3).
Graph the function h(x) = 1 / (x - 2) and indicate the vertical asymptote.

Remember, vertical asymptotes occur when the function is undefined, so they represent values to which the function cannot extend. Practice finding and graphing vertical asymptotes to become more familiar with this concept.