Honors Algebra II Supplemental Notes

Oblique Asymptotes
Key Concepts:
An oblique asymptote is an oblique, or slanted, line that the graph of a function approaches, but never touches, as x → ∞ ( x approaches infinity). To find the oblique asymptote of a rational function: 1. Reduce the rational function

P( x) to lowest terms. An oblique asymptote can be found Q( x) when the degree of P ( x) is one greater than the degree of Q ( x ) . c 2. Divide P ( x) by Q ( x ) using long division to get the form ax + b + , where a, b, and Q( x) c are constants. 3. The term

c c → 0 as x → ∞ . Eliminating the remainder , we are left Q( x) Q( x) with f ( x ) → ax + b as x → ∞ . y = ax + b is an oblique asymptote for the rational function P( x) . Q( x)

4.

f ( x) =

4 x3 + 9 x 2 x2 + x

Follow the steps outlined above to determine the oblique asymptote for the given rational function. The first step is to check that the rational function is reduced to lowest terms. If not, we must reduce it.

x(4 x 2 + 9 x) 4 x 2 + 9 x / f ( x) = = x( x + 1) x +1 /

4x + 5 x + 1 4x2 + 9x −(4 x 2 + 4 x)
5x −(5 x + 5)
−5
We use long division to put form ax + b +

4x2 + 9x in the x +1

c . Q ( x)

4x + 5 +

−5 x +1
Determine the value of f(x) as x → ∞ . This is the oblique asymptote.

f ( x) → 4 x + 5 y = 4x + 5

CTY Distance Math

Oblique Asymptotes

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y = f ( x) =

4x + 9x
3

x2 + x

y = 4x + 5

This graph displays both the rational expression f ( x) and its oblique asymptote. Note that in addition to the oblique asymptote, a vertical asymptote exists at x = −1 . These act as guidelines to shape the curve.

Practice Problems:
1. Find any oblique asymptotes for the rational function

2x2 + x + 1 . y = f ( x) = x
2. Find any oblique asymptotes for the rational function

2 x3 + x . y = f ( x) = 2 x −1
3. Find all of the asymptotes for the rational function

y = f ( x) =

2 x3 + x . ( x − 3)( x + 1)

4. Find all of the asymptotes for the rational function

y = f ( x) =

4 x3 + 5 x 2 + 6 . x 2 − 3x − 4

5. Find all of the asymptotes of the rational function

5x2 − 2 . y = f ( x) = x −3
Solutions:
1. y = 2 x + 1

y = 2x x = −1 and x = 3 are vertical asymptotes. y = 2 x + 4 is an oblique asymptote. 3. 4. x = −1 and x = 4 are vertical asymptotes. y = 4 x + 17 is an oblique asymptote. 5. x = 3 is a vertical asymptote. y = 5 x + 15 is an oblique asymptote.
2.

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Oblique Asymptotes

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