Sometimes it is not as easy to find the oblique asymptote. For example, the asymptote of
y
=
2
x
2
−
3
x
+
3
x
−
2
y equals . fraction 2 , x squared , minus 3 x plus 3 , over x minus 2 end fraction is not
2
x
2
x
fraction 2 , x squared , over x end fraction or
2
x
.
2 x . You can use polynomial division to find the oblique asymptote of any rational function.
Example 3
Determine the oblique asymptote of
y
=
2
x
2
−
3
x
+
3
x
−
2
.
y equals . fraction 2 , x squared , minus 3 x plus 3 , over x minus 2 end fraction . .
Divide the numerator by the denominator
.
x
−
2
2
x
+
1
2
x
2
−
3
x
+
3
2
x
2
−
4
x
_
x
+
3
x
−
2
_
5
Ignore the remainder
. Th e asymptote is the quotient
,
y
=
2
x
+
1.
table with 6 rows and 1 column , row1 column 1 , cap dividethenumeratorbythedenominator . . x minus 2 . table with 2 rows and 1 column , row1 column 1 , 2 x plus 1 , row2 column 1 , long division symbol enclosing , 2 , x squared , minus 3 x plus 3 , end symbol , end table , row2 column 1 , modified 2 , x squared , minus 4 x with under bar below , row3 column 1 , x plus 3 , row4 column 1 , modified x minus 2 with under bar below , row5 column 1 , 5 , row6 column 1 , cap ignoretheremainder . .cap theasymptoteisthequotient . comma y equals 2 x plus 1. , end table
Check
Use a graphing calculator to check your answer.
Image Long Description
Exercises
Graphing Calculator For each function determine the oblique asymptote. Check with a graphing calculator.
-
y
=
x
2
−
1
x
y equals . fraction x squared , minus 1 , over x end fraction
-
y
=
−
2
x
2
3
x
+
2
y equals negative . fraction 2 , x squared , over 3 x plus 2 end fraction
-
y
=
4
−
x
3
4
x
2
−
1
y equals . fraction 4 minus , x cubed , over 4 , x squared , minus 1 end fraction
-
y
=
2
x
4
+
99
,
999
x
3
y equals . fraction 2 , x to the fourth , plus 99 comma 999 , over x cubed end fraction
-
Technical Writing Write a step-by-step manual for classmates to use so they can use a spreadsheet to explore the differences between f(x) and g(x) as the value of x increases.
|
f
(
x
)
=
12
x
2
−
7
3
x
+
4
f , open x close , equals . fraction 12 , x squared , minus 7 , over 3 x plus 4 end fraction
|
g
(
x
)
=
4
x
g open x close equals 4 x
|
-
Open-Ended Write three rational functions that have an oblique asymptote of
y
=
2
x
.
y equals 2 x . Graph to check your work.
Describe the asymptotes of the graph of each function.
-
f
(
x
)
=
2
x
+
1
x
2
−
1
f , open x close , equals . fraction 2 x plus 1 , over x squared , minus 1 end fraction
-
f
(
x
)
=
x
2
−
9
x
+
3
f , open x close , equals . fraction x squared , minus 9 , over x plus 3 end fraction
-
f
(
x
)
=
5
x
+
11
4
x
+
6
f , open x close , equals . fraction 5 x plus 11 , over 4 x plus 6 end fraction
-
f
(
x
)
=
4
x
2
+
x
−
3
7
x
−
1
f , open x close , equals . fraction 4 , x squared , plus x minus 3 , over 7 x minus 1 end fraction
-
y
=
2
x
2
−
7
x
−
5
2
x
+
3
y equals . fraction 2 , x squared , minus 7 x minus 5 , over 2 x plus 3 end fraction
-
y
=
6
x
2
+
14
x
+
7
2
x
+
3
y equals . fraction 6 , x squared , plus 14 x plus 7 , over 2 x plus 3 end fraction
