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 is equal to 4, horizontal asymptote: y is equal to 2

vertical asymptote: x=−4
, horizontal asymptote: y=2
vertical asymptote: x is equal to negative 4, horizontal asymptote: y is equal to 2

vertical asymptote: y=2
, horizontal asymptote: x=−4
vertical asymptote: y is equal to 2, horizontal asymptote: x is equal to negative 4

vertical asymptote: x=4
, horizontal asymptote: y=−2
vertical asymptote: x is equal to 4, horizontal asymptote: y is equal to negative 2

Answer 1

To determine the vertical and horizontal asymptotes of the function \( y = \frac{3}{x + 4} + 2 \), let's analyze it step by step.

1. **Vertical Asymptote:**
- A vertical asymptote occurs where the function approaches infinity or negative infinity, which usually happens when the denominator of a fraction is zero.
- In this case, the vertical asymptote is found by setting the denominator to zero:
\[
x + 4 = 0 \implies x = -4
\]
- Therefore, the vertical asymptote is \( x = -4 \).

2. **Horizontal Asymptote:**
- To find the horizontal asymptote, we look at the behavior of the function as \( x \) approaches infinity (or negative infinity). For rational functions in the form \( \frac{a}{x + b} + c \), if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = c \).
- In this case:
\[
\lim_{x \to \infty} \left( \frac{3}{x + 4} + 2 \right) = \lim_{x \to \infty} \frac{3}{x + 4} + 2 = 0 + 2 = 2
\]
- So the horizontal asymptote is \( y = 2 \).

Putting it all together:

- Vertical asymptote: \( x = -4 \)
- Horizontal asymptote: \( y = 2 \)

Thus, the correct response is:
- **vertical asymptote: x is equal to -4, horizontal asymptote: y is equal to 2**.