A rational function is graphed on the coordinate plane.The graph consists of three curves. The first curve starts just above y = 2 and increases, approaching but never reaching x = negative 2. The second curve starts at the bottom of the graph just to the right of x = negative 2 and increases until just below y = 2, then decreases, approaching but never reaching x = 2. The third curve starts at the top of the graph just to the right of x = 2 and decreases down and to the right approaching but never reaching y = 2. There are arrows on both ends of each curve.

Question

Question
Use the graph to answer the question.
Identify all of the asymptotes for the graphed function. Select all that apply.
(2 points)
Responses

x = 2
x = 2

x = 0
x = 0

x equals negative 2
Image with alt text: x equals negative 2

y = 2
y = 2

y = 0
y = 0

y equals negative 2

Answer 1

Based on the description of the rational function graphed, the asymptotes can be identified as follows:

1. Vertical Asymptotes: These occur where the function approaches but never reaches a certain value of x. From the description:

   • The first curve approaches \( x = -2 \) but never touches it, indicating a vertical asymptote at \( x = -2 \).
   • The second curve approaches \( x = 2 \) but never touches it, indicating another vertical asymptote at \( x = 2 \).

2. Horizontal Asymptotes: These occur when the function approaches a specific value of y as \( x \) goes to infinity or negative infinity. From the description:

   • The curves approach \( y = 2 \) as they increase/decrease and move towards infinity. Therefore, there is a horizontal asymptote at \( y = 2 \).

Based on this analysis, the asymptotes for the graph of the rational function are:

• \( x = -2 \) (Vertical)
• \( x = 2 \) (Vertical)
• \( y = 2 \) (Horizontal)

The selected responses are:

• x = -2
• x = 2
• y = 2