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Slant Asymptotes

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

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Summary: Finding slant asymptotes.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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:

Exercise 1

y=3x3x2−1 y 3 x 3 x 2 1

Solution

y=3x y 3 x

Exercise 2

y=2x2x+1 y 2 x 2 x 1

Solution

y=2x−2 y 2 x 2

Exercise 3

y=x2−9x+2x+4 y x 2 9 x 2 x 4

Solution

y=x−13 y x 13

Exercise 4

y=x3−27x2+3 y x 3 27 x 2 3

Solution

y=x y x

Exercise 5

y=2x3+7x2−4x+3x−1 y 2 x 3 7 x 2 4 x 3 x 1

Solution

y=2x+3 y 2 x 3

Exercise 6

y=x2+5x+8x+3 y x 2 5 x 8 x 3

Solution

y=x+2 y x 2

Exercise 7

y=2x2+xx+1 y 2 x 2 x x 1

Solution

y=2x−1 y 2 x 1

Exercise 8

y=2xx+11x−4 y 2 x x 11 x 4

Solution

y=2x+30 y 2 x 30

Exercise 9

y=x4x−13 y x 4 x 1 3

Solution

y=x+3 y x 3

Exercise 10

y=x3−x+3x2+x−2 y x 3 x 3 x 2 x 2

Solution

y=x−1 y x 1

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