To identify the correct graph representing the rational function corresponding to the given table of values, we need to analyze the data. The important points and characteristics from the table are as follows:
- The function has a vertical asymptote at \( x = -2 \) because \( y \) is undefined at that point.
- We see defined values for \( y \) at various points leading up to the asymptote and after it, specifically showing two branches of the function.
- The points include:
- When \( x = -6 \), \( y \) approaches \( 0 \) from below.
- When \( x = -4 \), \( y \) is \( 5 \).
- When \( x = 0 \), \( y \) is \( -3 \).
- When \( x = 2 \), \( y \) is \( 2 \).
- When \( x = 4 \), \( y \) is \( 5 \).
Based on this analysis, the graph must indicate:
- A vertical dashed line at \( x = -2 \).
- A curve approaching this vertical asymptote from the left and continuing from the right.
- An open circle (indicating a hole) at \( (2, 2) \).
Among the available graph options, the correct one must include a vertical asymptote at \( x = -2 \) and must capture the behavior of the function as described in the points.
The first response describes the function with a dashed line at \( x = -2 \), which matches the required characteristics. It also states that the right branch has an open circle at \( (2, 2) \), which corresponds exactly to what we find in the data.
Thus, the correct graph is the first response: it has a vertical dashed line at \( x = -2 \) and reflects the behavior of the function as outlined by the table of values.
1 answer