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Rational Functions: Slant Asymptotes

Slant Asymptotes: A Slant asymptote of a rational function is a slant line (equation: y = mx + b) such that as values of the independent variable, x, decrease without bound or increase without bound, the function values (y-values) approach (get closer and closer to) the y-values of the points on the slant asymptote (from either above or below).

The next slide illustrates the definition.

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Rational Functions: Slant Asymptotes

Slide 2

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 1 2 3 4

slant asymptote (dashed line)

As x increases the y-values of points on the graph get closer to the y-values of points on the asymptote.

As x decreases the y-values of points on the graph get closer to the y-values of points on the asymptote.

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Rational Functions: Slant Asymptotes

Slide 3

Example: Find the slant asymptote of .12

3

x

xxf

Divide the numerator by the denominator using the "long division" process.

xx

xxx

3

32 1

- xThe slant asymptote is y = quotient.

slant asymptotey = x

(This was the function shown graphed in the preceding slide!)

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Rational Functions: Slant Asymptotes

Slide 4

Try: Find the slant asymptote of .1

32 2

xxx

xf

The slant asymptote is y = 2x + 3.

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Rational Functions: Slant Asymptotes