Slant Asymptotes
Summary: Finding slant asymptotes.
Just like vertical and horizontal asymptotes, slant asymptotes are lines the graph approaches. They are also called oblique asymptotes.
A graph has a slant asymptote if the degree of the numerator is bigger than the degree of the denominator (there is no horizontal asymptote).
To find slant asymptotes, divide the numerator by the denominator and keep only the quotient (the answer, throw away the remainder). Don't forget that these are still lines, so they are written as y =
To divide, you either have to use long division or synthetic division (if possible).
PRACTICE - Find the Slant Asymptotes:
Problem 1
y = 3 x 3 x 2 - 1 y
3
x
3
x
2
1
Solution 1
y = 3 x y
3
x
Problem 2
y = 2 x 2 x + 1 y
2
x
2
x
1
Solution 2
y = 2 x - 2 y
2
x
2
Problem 3
y = x 2 - 9 x + 2 x + 4 y
x
2
9
x
2
x
4
Solution 3
y = x - 13 y
x
13
Problem 4
y = x 3 - 27 x 2 + 3 y
x
3
27
x
2
3
Solution 4
y = x y
x
Problem 5
y = 2 x 3 + 7 x 2 - 4 x + 3 x - 1 y
2
x
3
7
x
2
4
x
3
x
1
Solution 5
y = 2 x + 3 y
2
x
3
Problem 6
y = x 2 + 5 x + 8 x + 3 y
x
2
5
x
8
x
3
Solution 6
y = x + 2 y
x
2
Problem 7
y = 2 x 2 + x x + 1 y
2
x
2
x
x
1
Solution 7
y = 2 x - 1 y
2
x
1
Problem 8
y = 2 x x + 11 x - 4 y
2
x
x
11
x
4
Solution 8
y = 2 x + 30 y
2
x
30
Problem 9
y = x 4 x - 1 3 y
x
4
x
1
3
Solution 9
y = x + 3 y
x
3
Problem 10
y = x 3 - x + 3 x 2 + x - 2 y
x
3
x
3
x
2
x
2
Solution 10
y = x - 1 y
x
1
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This work is licensed by Pradnya Bhawalkar and Kim Johnston. See the Creative Commons License about permission to reuse this material.
Last edited by Pradnya Bhawalkar on May 1, 2006 6:07 am GMT-5.