Question

Question 2
A)
The table of values represents a rational function.

x y
−6 0
−5 2
−4 5
−2 undefined
0 −3
2 2
4 5

Which graph shows the rational function?

(1 point)
Responses

The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 3 comma 6 right parenthesis and left parenthesis negative 3 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 3. The right branch starts to the right of the dashed line at x equals negative 3 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis.
Image with alt text: The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 3 comma 6 right parenthesis and left parenthesis negative 3 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 3. The right branch starts to the right of the dashed line at x equals negative 3 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis.

The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 2 comma 6 right parenthesis and left parenthesis negative 2 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 2. The right branch starts to the right of the dashed line at x equals negative 2 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis. An open circle is on the right branch of the curve at left parenthesis 2 comma 2 right parenthesis.
Image with alt text: The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 2 comma 6 right parenthesis and left parenthesis negative 2 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 2. The right branch starts to the right of the dashed line at x equals negative 2 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis. An open circle is on the right branch of the curve at left parenthesis 2 comma 2 right parenthesis.

The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 3 comma 6 right parenthesis and left parenthesis negative 3 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 3. The right branch starts to the right of the dashed line at x equals negative 3 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis. An open circle is on the right branch of the curve at 2, 2.
Image with alt text: The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 3 comma 6 right parenthesis and left parenthesis negative 3 comma negative 6 right parenthesis. The graph has two curved branches. The left branch starts curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 3. The right branch starts to the right of the dashed line at x equals negative 3 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis. An open circle is on the right branch of the curve at 2, 2.

The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 2 comma 6 right parenthesis and left parenthesis negative 2 comma negative 6 right parenthesis. The graph has two curved branches. The left branch curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 2. The right branch starts to the right of the dashed line at x equals negative 2 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis.
Image with alt text: The coordinate grid has its x axis labeled from negative 8 to 6 and y axis labeled from negative 6 to 6. The graph has a dashed line through left parenthesis negative 2 comma 6 right parenthesis and left parenthesis negative 2 comma negative 6 right parenthesis. The graph has two curved branches. The left branch curves up through left parenthesis negative 8 comma negative 3 right parenthesis; left parenthesis negative 6 comma 0 right parenthesis; and left parenthesis negative 4 comma 5 right parenthesis; and curves up as it approaches the dashed line at x equals negative 2. The right branch starts to the right of the dashed line at x equals negative 2 and curves up through left parenthesis 0 comma negative 3 right parenthesis; left parenthesis 1 comma 0 right parenthesis; left parenthesis 2 comma 2 right parenthesis; and left parenthesis 4 comma 5 right parenthesis.

pick an answer

Answer 1

To determine which graph represents the rational function based on the provided table of values, we need to pay close attention to the characteristics outlined:

- The function is undefined at \( x = -2 \).
- The values for \( y \) at various \( x \) values suggest that as \( x \) approaches \( -2 \), there is a discontinuity (i.e., an asymptote or hole).

The key points from the table are:
- At \( x = -6, y = 0 \)
- At \( x = -5, y = 2 \)
- At \( x = -4, y = 5 \)
- At \( x = -2, y = \) undefined (discontinuity)
- At \( x = 0, y = -3 \)
- At \( x = 2, y = 2 \)
- At \( x = 4, y = 5 \)

Based on the description of the graphs, we should choose the graph that has a vertical asymptote or a hole at \( x = -2 \) and shows the correct behavior around this point.

From the options given in your prompt, both graphs with a dashed line at \( x = -2 \) would imply a discontinuity at that point, which aligns with the data you've provided.

However, since the data suggest \( x = -2 \) is undefined in your provided values and the validated points describe behavior away from that discontinuity, I would choose the graph that emphasizes a break at that point, showing curves on both sides while respecting other listed points in the table.

Therefore, the correct graph is the one with a dashed line at \( x = -2 \), with the left branch approaching 5 at \( x = -4 \) and the right branch curving through the other points, aligning closely with the given values.

**Answer**: The graph with a dashed line through \( (−2, 6) \) and \( (−2, −6) \).