Special Graphs: Asymptotes and Holes

Special Graphs Asymptotes and Holes

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Asymptotes

An asymptote is a line that a graph approaches without touching.

If a graph has a horizontal asymptote of y = k, then part of the graph approaches the line y = k without touching it--y is almost equal to k, but y is never exactly equal to k. The following graph has a horizontal asymptote of y = 3:

The following graph has a horizontal asymptote of y = 0:

If a graph has a vertical asymptote of x = h, then part of the graph approaches the line x = h without touching it--x is almost equal to h, but x is never exactly equal to h. The following graph has a vertical asymptote of x = 3:

One reason vertical asymptotes occur is due to a zero in the denominator of a rational function. For example, if f (x) = , then x cannot equal 5, but x can equal values very close to 5 (4.99, for example). The graph of f (x) = looks like:

Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value. In the previous graph, there is no value of x for which y = 0 (≠ 0), but as x gets very large or very small, y comes close to 0. Thus, f (x) = has a horizontal asymptote at y = 0.

The graph of a function may have several vertical asymptotes. f (x) = has vertical asymptotes of x = 2 and x = - 3, and f (x) = has vertical asymptotes of x = - 4 and x = . In general, a vertical asymptote occurs in a rational function at any value of x for which the denominator is equal to 0, but for which the numerator is not equal to 0.