Special Graphs
Asymptotes and Holes
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
:
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.
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
:
Examples: Name the vertical asymptotes and holes in the graphs of the
following equations:
-
f (x) =
-
f (x) =
-
f (x) =
Solutions:
- Asymptotes: x = 3, - 3 . Holes: x = - 4 .
- Asymptotes: x = 2 . Holes: x = 0 .
- Asymptotes:
x = - 1,
, - 12
. Holes: NONE.
