Question

Describe the vertical asymptote(s) and hole(s) for the graph of y = (x−3)(x+8)/(x+8)(x+2)
(1 point)
Responses

asymptote: x=−8 and hole: x=−2


asymptote: x=2 and hole: x=8



asymptote: x=−2 and hole: none


asymptote: x=−2 and hole: x=−8

Answer 1

To analyze the function \( y = \frac{(x-3)(x+8)}{(x+8)(x+2)} \) for vertical asymptotes and holes, we first need to find the points where the function is undefined, which typically occur at the roots of the denominator.

1. **Identify the factors in the denominator:**
The denominator is \( (x+8)(x+2) \).
This means the function is undefined at \( x = -8 \) and \( x = -2 \).

2. **Check for common factors:**
The numerator is \( (x-3)(x+8) \). Notice that \( (x + 8) \) is a common factor in both the numerator and the denominator. This indicates that there is a hole at \( x = -8 \) since it will cancel out.

3. **Determine vertical asymptotes:**
After cancelling the common factor, we rewrite the function, focusing on the remaining part:
\[ y = \frac{x - 3}{x + 2} \]
The remaining denominator \( x + 2 \) will go to zero (and thus create a vertical asymptote) when \( x = -2 \).

4. **Conclusion:**
- **Hole:** Since \( (x + 8) \) canceled out, there is a hole at \( x = -8 \).
- **Vertical asymptote:** The function has a vertical asymptote at \( x = -2 \).

Based on this analysis, the correct response is:

**asymptote: x=−2 and hole: x=−8**.