Question

What are the asymptotes for y = 3/x+4 + 2?
(1 point)
Responses

vertical asymptote: x=4, horizontal asymptote: y=2


vertical asymptote: x=−4, horizontal asymptote: y=2


vertical asymptote: y=2, horizontal asymptote: x=−4


vertical asymptote: x=4, horizontal asymptote: y=−2

Answer 1

To find the asymptotes for the function \( y = \frac{3}{x + 4} + 2 \), we can analyze the function step by step.

1. **Vertical Asymptote**: This occurs where the denominator is zero (as this leads to the function approaching infinity). For the function \( \frac{3}{x + 4} \), set the denominator equal to zero:
\[
x + 4 = 0 \implies x = -4
\]
So, there is a vertical asymptote at \( x = -4 \).

2. **Horizontal Asymptote**: This is determined by the end behavior of the function as \( x \) approaches infinity or negative infinity. As \( x \) approaches infinity (or negative infinity), the term \( \frac{3}{x + 4} \) approaches 0. Therefore, the function approaches:
\[
y = 0 + 2 = 2
\]
So, the horizontal asymptote is \( y = 2 \).

Putting it all together, we find:
- **Vertical Asymptote**: \( x = -4 \)
- **Horizontal Asymptote**: \( y = 2 \)

Thus, the correct response is:
**vertical asymptote: x=−4, horizontal asymptote: y=2**.