To find the $y$-intercept of the rational function:

$f(x) = \frac{x + 9}{x^2 + 6x - 27}$

we need to evaluate $f(0)$, since the $y$-intercept occurs when $x = 0$.

1. Substitute $x = 0$ into the function:

   $f(0) = \frac{0 + 9}{0^2 + 6 \cdot 0 - 27} = \frac{9}{-27} = -\frac{1}{3}$

So, the $y$-intercept of the function is $-\frac{1}{3}$.

The correct answer is:

• $-\frac{1}{3}$

Would you like further details or have any questions about this solution?

Here are five related questions to expand your understanding:

1. How do you find the $x$-intercepts of a rational function?
2. What is the domain of the function $f(x) = \frac{x + 9}{x^2 + 6x - 27}$?
3. How can we determine the vertical asymptotes of a rational function?
4. What are horizontal asymptotes, and how do you find them in rational functions?
5. What happens to the $y$-intercept if the function is shifted vertically?

Tip: Always check for any factors that might simplify a rational function before finding intercepts or asymptotes.