The function f(x) = x^2 + 2x - 15/x^2 + 9x + 20 has a vertical asymptote at x = –5.
Question 16 options:
True
False

To determine if the function \( f(x) = \frac{x^2 + 2x - 15}{x^2 + 9x + 20} \) has a vertical asymptote at \( x = -5 \), we need to analyze the denominator of the function, as vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.

First, factorize the denominator:

\[ x^2 + 9x + 20 = (x + 4)(x + 5) \]

The denominator is zero when:

\[ (x + 4)(x + 5) = 0 \]

This occurs at \( x = -4 \) and \( x = -5 \).

Next, we check the numerator at these points to ensure they do not simultaneously equal zero. Factorize the numerator:

\[ x^2 + 2x - 15 = (x + 5)(x - 3) \]

The numerator is zero when:

\[ (x + 5)(x - 3) = 0 \]

This occurs at \( x = -5 \) and \( x = 3 \).

Since \( x = -5 \) makes both the numerator and denominator zero, we need to check if it’s a removable discontinuity or a vertical asymptote. If the factor \((x + 5)\) can be canceled from both the numerator and the denominator, then \( x = -5 \) is a removable discontinuity.

\[ f(x) = \frac{(x + 5)(x - 3)}{(x + 5)(x + 4)} \]

Cancel the common factor \((x + 5)\):

\[ f(x) = \frac{x - 3}{x + 4} \]

So, at \( x = -5 \), the function has a hole (removable discontinuity) rather than a vertical asymptote.

Thus, the statement that the function \( f(x) \) has a vertical asymptote at \( x = -5 \) is:

**False**

Would you like more details or have any questions?

Here are 8 related questions:
1. What are the vertical asymptotes of the function \( f(x) = \frac{x - 3}{x + 4} \)?
2. How do you determine the horizontal asymptote of a rational function?
3. What is the difference between a vertical asymptote and a hole in a graph?
4. Can a function have more than one vertical asymptote?
5. How do you find the x-intercepts of the function \( f(x) = \frac{x - 3}{x + 4} \)?
6. How do you find the y-intercept of the function \( f(x) = \frac{x - 3}{x + 4} \)?
7. What is the end behavior of the function \( f(x) = \frac{x - 3}{x + 4} \)?
8. How does the factorization of the numerator and denominator help in analyzing the graph of a rational function?

**Tip:** Always check for common factors in the numerator and denominator when analyzing rational functions for vertical asymptotes and holes.

Explore the concept of vertical asymptotes in rational functions through an example analysis of the function f(x) = (x^2 + 2x - 15) / (x^2 + 9x + 20). Discover how factorization of numerator and denominator helps determine the presence of vertical asymptotes. Suitable for advanced high school students.

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