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Vertical Asymptotes (page 1 of 4) Sections: Vertical asymptotes, Horizontal asymptotes, Slant asymptotes, Examples 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: This is a rational function. More to the point, this is a fraction. Can you 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 cannot be: x^{2}
– 5x – 6 = 0
So x cannot be 6 or –1, because then I'd be dividing by zero.
You can see how the graph avoided the vertical lines x = 6 and x = –1. This avoidance occurred because x cannot be –1 or 6. In other words, the fact that the function's domain is restricted is reflected in the function's graph. More usefully, you can use the domain to help you graph, because whichever values are not allowed in the domain will be vertical asymptotes on the graph.
Let's review this relationship between the domain and the vertical asymptotes.
The domain is the set of all xvalues 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
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:
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 is always 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
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 xvalues, and the domain is "all x". Also, since there are no values forbidden to the domain, there are no vertical asymptotes. domain:
all x
Note again how the domain and vertical asymptotes were "opposites" of each other.
I'll check the zeroes of the denominator: x^{2}
+ 5x + 6 = 0
Since I can't divide by zero, then I have vertical asymptotes at x = –3 and x = –2, and the domain is all other xvalues. domain:
When graphing, remember that vertical asymptotes stand for xvalues that are not allowed. Vertical asymptotes are sacred ground. Never, on pain of death, can you cross a vertical asymptote. Don't even try! Top  1  2  3  4  Return to Index Next >>



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