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# Rational Functions Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

#### Examples

• From above, we know that the x -axis, or the horizontal line y = 0, is a horizontal asymptote. Here is why. Because the highest power on x is 4, we will multiply the fraction by which reduces to 1, so we are not changing the fraction.

For large values of x , 3/ x , 5/ x 2 , 1/ x 3 , 6/ x 4 , 8/ x 2 , and 1/ x 4 are very close to zero, so for large values of x ,

•

• The degree of the numerator equals the degree of the denominator, so the graph of this function has a horizontal asymptote at the line y = 4/9. Here is why. Because the largest power on x is 3, we will multiply the fraction by .

For large values of is close to .

These steps are not necessary to find the horizontal asymptotes, only the three rules earlier in this guide.

## Rational Functions Practice Problems

#### Practice

Find the intercepts, vertical asymptotes, and horizontal asymptotes.

#### Solutions

1. The vertical is asymptote , from 2 x + 3 = 0. The horizontal asymptote is because the numerator and denominator have the same degree. The x -intercept is −2, from x + 2 = 0. The y -intercept is

2. The vertical asymptotes are x = − 5 and x = 4, from x 2 + x − 20 = 0. The horizontal asymptote is y = 0 because the denominator has the higher degree. The x -intercept is 0, from −3 x = 0. The y -intercept is

3. There is no vertical asymptote because x 2 + 1 = 0 has no real solution. The horizontal asymptote is y = 1/1 = 1 because the numerator and denominator have the same degree. The x -intercepts are ±1, from x 2 − 1 = 0. The y -intercept is

4. The vertical asymptote is from 8 x + 3 = 0. There is no horizontal asymptote because the numerator has the higher degree. The x -intercepts are , from 9 x 2 − 1 = 0. The y -intercept is

5. There is no vertical asymptote because x 2 + 4 = 0 has no real solution. There is no horizontal asymptote because the numerator has the higher degree. The x -intercept is −1, from x 3 + 1 = 0. The y -intercept is

6. The vertical asymptote is x = 0, from x 2 = 0. The horizontal asymptote y = 0 because the denominator has the higher degree. There is no x -intercept because the numerator is 2, never 0. There is no y -intercept because 2/0 2 is not defined.

Practice problems for this concept can be found at: Rational Functions Practice Test.

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