Question
The table of values represents a rational function.x y−6 0−5 2−4 5−2 undefined0 −3 2 24 5Which graph shows the rational function?(1 point)ResponsesImage 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.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.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.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.Graphing CalculatorSkip to navigationPreviousNextpage 20 of 20
Answers
GPT-4o mini
To identify the correct graph representing the rational function based on the table of values provided, we first analyze the given values for x and y:
- When \( x = -6 \), \( y = 0 \)
- When \( x = -5 \), \( y = 2 \)
- When \( x = -4 \), \( y = 5 \)
- When \( x = -3 \), \( y \) is undefined
- When \( x = -2 \), \( y = 24 \)
- When \( x = 0 \), \( y = -3 \)
- When \( x = 2 \), \( y = 2 \)
- When \( x = 4 \), \( y = 5 \)
From this data, we can observe the following characteristics:
1. The function is undefined at \( x = -3 \), indicating a vertical asymptote at that x-value.
2. The function has values approaching infinity as \( x \) approaches \(-3\) from the left and from the right.
3. The function exhibits a noticeable increase in value between the points where \( x = -2 \) (yielding a large positive y value) and \( x = 0 \) (yielding a negative y value).
Now, evaluating the graph options based on these characteristics:
- The correct graph must show a vertical asymptote at \( x = -3 \) (not at \( x = -2\)).
- It should have two branches; one should approach the asymptote from the left while going to infinity, and the other should approach the asymptote from the right while starting from low values and increasing.
Based on these observations, we can conclude that the appropriate graph is the one that has:
- A vertical dashed line at \( x = -3 \).
- Curved branches on both sides of this line.
- The left branch approaching \( x = -3 \) from below and the right branch beginning after \( x = -3 \) and increasing.
Thus, the correct response is the option that shows a vertical asymptote at \( x = -3 \).
- When \( x = -6 \), \( y = 0 \)
- When \( x = -5 \), \( y = 2 \)
- When \( x = -4 \), \( y = 5 \)
- When \( x = -3 \), \( y \) is undefined
- When \( x = -2 \), \( y = 24 \)
- When \( x = 0 \), \( y = -3 \)
- When \( x = 2 \), \( y = 2 \)
- When \( x = 4 \), \( y = 5 \)
From this data, we can observe the following characteristics:
1. The function is undefined at \( x = -3 \), indicating a vertical asymptote at that x-value.
2. The function has values approaching infinity as \( x \) approaches \(-3\) from the left and from the right.
3. The function exhibits a noticeable increase in value between the points where \( x = -2 \) (yielding a large positive y value) and \( x = 0 \) (yielding a negative y value).
Now, evaluating the graph options based on these characteristics:
- The correct graph must show a vertical asymptote at \( x = -3 \) (not at \( x = -2\)).
- It should have two branches; one should approach the asymptote from the left while going to infinity, and the other should approach the asymptote from the right while starting from low values and increasing.
Based on these observations, we can conclude that the appropriate graph is the one that has:
- A vertical dashed line at \( x = -3 \).
- Curved branches on both sides of this line.
- The left branch approaching \( x = -3 \) from below and the right branch beginning after \( x = -3 \) and increasing.
Thus, the correct response is the option that shows a vertical asymptote at \( x = -3 \).