Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.)
Let's consider the following equation:
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This is a rational function. More to the point, this is a fraction. Can we have a zero in the denominator of a fraction? No. 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:
x^{2} – 5x – 6 = 0
(x – 6)(x + 1) = 0
x = 6 or –1
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.
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 the side" aspect is another part of the definition of a vertical asymptote. 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.
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Let's do some practice with this relationship between the domain of the function and its vertical asymptotes.
The domain is the set of all x-values that I'm allowed to use. The only values that could be disallowed are those that give me a zero in the denominator. So I'll set the denominator equal to zero and solve.
x^{2} + 2x – 8 = 0
(x + 4)(x – 2) = 0
x = –4 or x = 2
Since I can't have a zero in the denominator, then I can't have x = –4 or x = 2 in the domain. This tells me that the vertical asymptotes (which tell me where the graph can not go) will be at the values x = –4 or x = 2.
domain: x ≠ –4, 2
vertical asymptotes: x = –4, 2
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.
To find the domain and vertical asymptotes, I'll set the denominator equal to zero and solve. The solutions will be the values that are not allowed in the domain, and will also be the vertical asymptotes.
x^{2} + 9 = 0
x^{2} = –9
Oops! That doesn't solve! So there are no zeroes in the denominator. Since there are no zeroes in the denominator, then there are no forbidden x-values, and the domain is "all x". Also, since there are no values forbidden to the domain, there are no vertical asymptotes.
domain: all x
vertical asymptotes: none
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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".
I'll check the zeroes of the denominator:
x^{2} + 5x + 6 = 0
(x + 3)(x + 2) = 0
x = –3 or x = –2
Since I can't divide by zero, then I have vertical asymptotes at x = –3 and x = –2, and the domain is all other x-values.
domain: x ≠ –3, –2
vertical asymptotes: x = –3, –2
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!
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