2.
A rational function is a quotient of
two polynomial functions.
)(
)(
)(
xh
xg
xf 0)( xh
The parent function is
,
x
xf
1
)( 0x
3.
I
III
a vertical asymptote x = 0
a horizontal
asymptote
y = 0
4.
Example 1.
Determine the asymptotes for the graph
of f(x) = .
1
34
x
x
A vertical asymptote
Solve x +1 = 0
x = -1
Since f(-1) is undefined, there may be a
vertical asymptote at x = -1
5.
A horizontal asymptote
1
34
)(
x
x
xf
1
34
x
x
y
Solve for xy(x + 1) = 4x + 3
xy + y = 4x + 3
xy - 4x = 3 - y
x(y - 4) = 3 - y
4
3
y
y
x
The rational expression is undefined
for y = 4. Thus, the horizontal
asymptote is the line y = 4.
6.
The graph of this function verifies that
the lines of y = 4 and x = -1 are asymptotes.
f(x) =
1
34
x
x
7.
Here’s the cheat sheet!
a) If the degree of the numerator is greater
than the degree of the denominator then
there is no a horizontal asymptote.
63x
2x
y
2
8.
b). If the degree of the numerator is less than
the degree of the denominator then a horizontal
asymptote is y=0
1x
14x
y 2
9.
c). If the degree of the numerator is equal to
the degree of the denominator then a horizontal
asymptote is a quotient.
16x
4x
y 2
2
4
x
4x
y 2
2
a horizontal asymptote
10.
Example 2
Use the parent graph
to graph each function. Describe the transformation(s)
that take place. Identify the new location of each
asymptote.
x
1
f(x)
a. g(x) =
To graph g(x) translate the parent
graph 2 units to the right. The new
vertical asymptote is x = 2. The
horizontal asymptote, y = 0, remains
unchanged.
2-x
1
11.
b. h(x) = 3x
1
To graph h(x) , reflect the parent graph over the
x-axis, and compress the result horizontally by a
factor of 3. This does not affect the vertical
asymptote at x = 0. The horizontal asymptote,
y = 0, is also unchanged.
12.
To graph m(x) , stretch the parent function
vertically by a factor of 3, and translate the
result 1 unit to the right and 2 units down. The
new vertical asymptote is x = 1. The horizontal
asymptote changes from y = 0 to
y = -2.
2
1x
3
m(x)
c.
13.
A slant asymptote occurs when the
degree of the numerator of a rational
function is exactly one degree greater
then the degree of a denominator.
14.
Example 3
Determine the slant asymptote for
1x
42x3x
xf
2
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