To determine the vertical and horizontal asymptotes of the rational function:

$f(x) = \frac{6x + 13}{2x + 9}$

Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero (since the function is undefined there). We solve:

$2x + 9 = 0$

$x = -\frac{9}{2}$

Thus, there is a vertical asymptote at $x = -\frac{9}{2}$.

Horizontal Asymptotes

To find horizontal asymptotes, compare the degrees of the numerator and the denominator:

• The degree of the numerator (6x + 13) is 1.
• The degree of the denominator (2x + 9) is also 1.

When the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients:

$y = \frac{6}{2} = 3$

Thus, the horizontal asymptote is $y = 3$.

Summary

• Vertical Asymptote: $x = -\frac{9}{2}$
• Horizontal Asymptote: $y = 3$

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Follow-Up Questions

1. How do you determine the vertical asymptote for a more complex rational function?
2. What happens if the degree of the numerator is greater than the degree of the denominator?
3. How can you find the slant asymptote if it exists?
4. What is the behavior of the function as $x$ approaches the asymptotes?
5. How do you interpret horizontal asymptotes in terms of the function's end behavior?

Tip

Remember, a horizontal asymptote indicates the behavior of a function as $x$ approaches infinity, while a vertical asymptote shows where the function is undefined.