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
Image with alt text: y equals negative 2

Answer 1

Based on the description of the rational function and its behavior, we can identify the asymptotes as follows:

1. Vertical Asymptotes: These occur where the function approaches infinity or negative infinity as it nears certain x-values (which are not in the domain of the function).

   • The graph approaches but never reaches \( x = -2 \). Therefore, there is a vertical asymptote at:

     • \( x = -2 \)

   • The graph also approaches but never reaches \( x = 2 \). Thus, there is another vertical asymptote at:

     • \( x = 2 \)

2. Horizontal Asymptotes: These are identified where the function approaches a constant value as \( x \) approaches infinity or negative infinity.

   • The graph approaches but never reaches \( y = 2 \). Therefore, there is a horizontal asymptote at:
     • \( y = 2 \)

Now let's summarize the asymptotes for the graphed function:

• Vertical Asymptotes: \( x = -2 \) and \( x = 2 \)
• Horizontal Asymptote: \( y = 2 \)

The correct responses to select would be:

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

Therefore, the identified asymptotes are:

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