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Rational Functions: Vertical Asymptotes

Vertical Asymptotes: A vertical asymptote of a rational function is a vertical line (equation: x = number) such that as values of the independent variable, x, approach (get closer and closer to) the number (from either the left or right side of the number), the function values (y-values) decrease without bound or increase without bound.

The next two slides illustrate the definition.

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Rational Functions: Vertical Asymptotes

Slide 2

The rational function shown graphed has vertical asymptote: x = 3, since as x-values approach 3 from the "left" of 3, the y values decrease without bound.

-30

-20

-10

0

10

20

30

-2 2 4 6

x 2.7 2.8 2.9 2.99

y -18 -28 -58 -598

x approaching 3 from the lefty decreasing without bound

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Rational Functions: Vertical Asymptotes

Slide 3

Also note that as x-values approach 3 from the "right" of 3, the y values increase without bound.

-30

-20

-10

0

10

20

30

-2 2 4 6

x 3.3 3.2 3.1 3.01

y 22 32 62 602

x approaching 3 from the righty increasing without bound

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Rational Functions: Vertical Asymptotes

Slide 4

This function’s only vertical asymptote is x = 3.(This was the function shown graphed in the preceding slides!)

Next, set the denominator equal to zero and solve.

x – 3 = 0x = 3

Example 1: Algebraically find the vertical asymptotes of,

.3

2

x

xxf

First, make sure the rational function can not be simplified further.

In this case, f (x), is simplified.

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Rational Functions: Vertical Asymptotes

Slide 5

This function’s vertical asymptotes are x = - 2 and x = 1.

Try: Algebraically find the vertical asymptotes of,

.2

32

xx

xxf

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Rational Functions: Vertical Asymptotes