Asked by burnout
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 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.
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 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.
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.
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 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.
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 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.
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.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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:
1. The function has a **vertical asymptote** at \( x = -2 \) because \( y \) is undefined at that point.
2. We see defined values for \( y \) at various points leading up to the asymptote and after it, specifically showing two branches of the function.
3. 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. The function has a **vertical asymptote** at \( x = -2 \) because \( y \) is undefined at that point.
2. We see defined values for \( y \) at various points leading up to the asymptote and after it, specifically showing two branches of the function.
3. 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.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.