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 is equal to negative 8 and hole: x is equal to negative 2

asymptote: x=2
and hole: x=8

asymptote: x is equal to 2 and hole: x is equal to 8

asymptote: x=−2
and hole: none
asymptote: x is equal to negative 2 and hole: none

asymptote: x=−2
and hole: x=−8

Answer 1

To analyze the rational function \( y = \frac{(x - 3)(x + 8)}{(x + 8)(x + 2)} \), we need to look for vertical asymptotes and holes.

1. **Identify holes**: A hole occurs where a factor in the numerator and denominator cancel each other out. In this case, the factor \( (x + 8) \) is present in both the numerator and the denominator. Thus, there is a hole at:
\[
x + 8 = 0 \implies x = -8
\]

2. **Identify vertical asymptotes**: Vertical asymptotes occur where the denominator is equal to zero, but the corresponding factor in the numerator does not cancel it out. The remaining factors in the denominator are \( (x + 2) \):
\[
x + 2 = 0 \implies x = -2
\]
Here, there is no corresponding factor in the numerator, so we have a vertical asymptote at \( x = -2 \).

Putting this together, we find:

- Hole: \( x = -8 \)
- Vertical asymptote: \( x = -2 \)

Thus, the correct answer would be:

**asymptote: x=−2 and hole: x=−8**.