Vertical Asymptotes

When graphing rational functions, we draw dashed vertical lines indicating these no-go portions of the graphing plane.

(Vertical asymptotes can also arise in other contexts, such as logarithms or tangent functions, but you'll almost certainly first encounter asymptotes in the context of rational functions.)

What is an example of finding a vertical asymptote?

Consider the following rational function:

This is a polynomial fraction, and we cannot have zeroes in fractions' denominators. So if I set the denominator of the above fraction equal to zero and solve, this will tell me the values that x can *not* be:

So x cannot be 6 or −1, because then I'd be dividing by zero.

Now let's look at the graph of this rational function:

You can see how the graph avoided the vertical lines x = 6 and x = −1. This avoidance occurred because x cannot be equal to either −1 or 6. In other words, the fact that the function's domain is restricted is reflected in the function's graph.

To be clear, this rational function has two vertical asymptotes, being the vertical lines x = −1 and x = 6.

We draw the vertical asymptotes as dashed lines to remind us not to graph there, like this:

It's alright that the graph appears to climb right up the sides of the asymptote on the left. This is common. As long as you don't draw the graph crossing the vertical asymptote, you'll be fine.

In fact, this "crawling up (or down) the side" aspect is another part of the definition of a vertical asymptote: the graph getting as close as you like to that vertical line, but without ever actually touching it.

We'll later see an example of where a zero in the denominator doesn't lead to the graph climbing up or down the side of a vertical line. But for now, and in most cases, zeroes of the denominator will lead to vertical dashed lines and graphs that skinny up as close as you please to those vertical lines.

Let's do some practice with this relationship between the domain of the function and its vertical asymptotes.

• Find the domain and vertical asymptotes(s), if any, of the following function:

Note that the domain and vertical asymptotes are "opposites". The vertical asymptotes are at −4 and 2, and the domain is everywhere but −4 and 2. This relationship always holds true.

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• Find the domain and vertical asymptote(s), if any, of the following function:

Note again how the domain and vertical asymptotes were "opposites" of each other. The domain is "all x-values" or "all real numbers" or "everywhere" (these all being common ways of saying the same thing), while the vertical asymptotes are "none".

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• Find the domain and vertical asymptote(s), if any, of the following function:

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When graphing, remember that vertical asymptotes stand for x-values that are not allowed. Vertical asymptotes are sacred ground. Never, on pain of death, can you cross a vertical asymptote. Don't even try!