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Special Graphs

Asymptotes and Holes

Problems

Problems

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 :

Horizontal Asymptote y = 3
The following graph has a horizontal asymptote of y = 0 :
Horizontal Asymptote 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 :

Vertical Asymptote 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:

f (x) =

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.

Holes

When a value of x sets both the denominator and the numerator of a rational function equal to 0, there is a hole in the graph; that is, a single point at which the function has no value. f (x) = has a hole at x = 2 :

f (x) =


Examples: Name the vertical asymptotes and holes in the graphs of the following equations:

  1. f (x) =
  2. f (x) =
  3. f (x) =

Solutions:
  1. Asymptotes: x = 3, - 3 . Holes: x = - 4 .
  2. Asymptotes: x = 2 . Holes: x = 0 .
  3. Asymptotes: x = - 1,, - 12 . Holes: NONE.

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