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To solve this problem, let's analyze the rational function given:

\[
f(x) = \frac{x^3 - 8}{-x^3 - 8x^2}
\]

We need to answer four questions: finding the x-intercepts, y-intercepts, vertical asymptotes, and horizontal asymptotes. Let's go through each part step-by-step.

### Part (a): X-intercepts
The x-intercepts occur where \( f(x) = 0 \). This means the numerator of \( f(x) \) must be zero (since a rational function equals zero when the numerator is zero, and the denominator is not zero).

1. **Set the numerator equal to zero:**
   \[
   x^3 - 8 = 0
   \]
2. **Solve for \( x \):**
   \[
   x^3 = 8
   \]
   \[
   x = \sqrt[3]{8} = 2
   \]
   
So, the x-intercept is at \( x = 2 \).

### Part (b): Y-intercepts
The y-intercept occurs where \( x = 0 \).

1. Substitute \( x = 0 \) into \( f(x) \):
   \[
   f(0) = \frac{0^3 - 8}{-0^3 - 8 \cdot 0^2} = \frac{-8}{0}
   \]

Since this results in division by zero, there is **no y-intercept**.

### Part (c): Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero (and the numerator is not zero).

1. **Set the denominator equal to zero:**
   \[
   -x^3 - 8x^2 = 0
   \]
   Factor out \( -x^2 \):
   \[
   -x^2(x + 8) = 0
   \]
2. **Solve for \( x \):**
   \[
   x^2 = 0 \quad \text{or} \quad x + 8 = 0
   \]
   This gives \( x = 0 \) or \( x = -8 \).

So, the vertical asymptotes are at \( x = 0 \) and \( x = -8 \).

### Part (d): Horizontal Asymptotes
To find the horizontal asymptote, we compare the degrees of the numerator and denominator.

- The numerator \( x^3 - 8 \) has degree 3.
- The denominator \( -x^3 - 8x^2 \) also has degree 3.

Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.

\[
y = \frac{1}{-1} = -1
\]

Thus, the horizontal asymptote is \( y = -1 \).

### Final Answers
- (a) The function has x-intercept(s) at \( x = 2 \).
- (b) The function has y-intercept(s) at \( y = \text{NONE} \).
- (c) The function has vertical asymptote(s) when \( x = 0, -8 \).
- (d) The function has horizontal asymptote(s) when \( y = -1 \).

Would you like further details or have any questions?

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Here are some related questions to expand on this topic:

1. How do you find slant (oblique) asymptotes for rational functions?
2. What happens to the graph of a function near a vertical asymptote?
3. How would the function change if the degrees of the numerator and denominator were different?
4. What is the significance of the x-intercepts in graphing a rational function?
5. How do holes in the graph of a rational function differ from vertical asymptotes?

**Tip:** When finding asymptotes, always start by simplifying the rational function, as factoring might reveal removable discontinuities.

Find all zeros and vertical asymptotes of the rational function f(x) = (x^3 - 8) / (-x^3 - 8x^2).

Solve a rational function problem involving finding zeros, vertical asymptotes, and horizontal asymptotes. Learn how to find x- and y-intercepts, vertical asymptotes, and horizontal asymptotes of f(x) = (x^3 - 8) / (-x^3 - 8x^2), covering essential algebra concepts.

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