Vertical asymptotes are interesting mathematical phenomena that occur with certain functions. You will learn where you will see them, what they look like, and how to find them in this lesson.
What Are Vertical Asymptotes?
Vertical asymptotes are invisible vertical lines that certain functions approach, yet do not cross, when the function is graphed. When you graph some mathematical functions, you will see that the resultant curve avoids certain invisible lines in the graph. No matter what, you can’t get the graph to cross those lines. Let me show you what it looks like.
The dashed lines have been drawn in to show you where the vertical asymptotes are. Do you see how the graph avoids those areas?
There are some rules that vertical asymptotes follow.
- The graph tends to either positive or negative infinity as it gets closer to the vertical asymptote. Look at the graph and notice how the curve goes either all the way up or all the way down as it nears the asymptote.
- The distance between the asymptote and the graph tends to zero as the graph gets closer to the asymptote. The graph and the asymptote will seem to almost merge together at the tips, but the curve will never actually touch the asymptote. It is as if the vertical asymptote had a protective field around it preventing anything from touching or crossing it.
- The graph can approach the vertical asymptote from either direction, from either the right or the left. Look at the graph and see how the graph approaches from both directions. Some functions only approach from only one direction, but like our function, others can approach from both.
The function that we graphed is somewhat complex and is called a rational function. In this lesson, we will focus on the vertical asymptotes of rational functions. There are other functions that also produce vertical asymptotes, but rational functions are the most common.
A rational function is a function whose numerator and denominator are made up of polynomials. The general form of a rational function is the following.
Here are some examples of rational functions.
All of the above are fractions where both the numerator and denominator are polynomials. Because of this, this type of function makes it easy for you to find the vertical asymptotes.
Determining Vertical Asymptotes
To determine the vertical asymptotes of a rational function, all you need to do is to set the denominator equal to zero and solve. Vertical asymptotes occur where the denominator is zero. Remember, division by zero is a no-no. Because you can’t have division by zero, the resultant graph thus avoids those areas.
Let’s go back to our first function and see if we can find the vertical asymptotes.
To find the vertical asymptotes, you need to set the denominator equal to zero and solve. Let’s see what we get when we do that. We would use factoring to solve.
We have found that our zeroes for our denominator are -3 and -7. Now, look at the graph to see if that is where my vertical asymptotes are. Yep, looks like it. The graph avoids the lines at x=-3 and x=-7.
There is one circumstance where a zero in the denominator does not produce a vertical asymptote. This is when you have the same zero in the numerator. So, what this means is that you would want to solve both the numerator and denominator for zero. If they have an answer in common, then that number is not a vertical asymptote. Let’s see what that looks like. The following function has already been factored, so you can easily see your zeroes.
Looking at this function, we see that the vertical asymptotes are -3, -1, and -2 from solving the denominator for zero. But, solving the numerator for zero, we see that the numerator has zeroes of -3 and 4. They both have a -3, so that means the vertical asymptote at -3 is canceled by the -3 zero in the numerator. So, my actual asymptotes are only x=-1 and x=-2.
To recap, a vertical asymptote is an invisible line which the graph never touches. The graph will approach this line, but it won’t dare touch or cross it. The graph can approach this asymptote from either direction – or both. To find the asymptote of rational functions, you solve the denominator for zero. All the zeroes of the denominator are vertical asymptotes, except in the case where the same zero occurs in the numerator.