The table of values represents a rational function.

Question

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

option 1: 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.

option 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 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.

option 3: 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.

option 4: 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.

Answer 1

To determine which graph represents the rational function based on the given values, we need to analyze the information.

1. Identify where y is undefined:

   • In the provided table of values, y is undefined when x = -2. This means there is a vertical asymptote at x = -2 for the rational function.

2. Examine the other points:

   • The points (−6, y), (−5, 2), (−4, 5), (−2, undefined), (0, −3), (2, 2), (4, 5) suggest that we need both branches of the function to curve through some of these points:
     • For x < -2: There are values such as (−6, y), (−5, 2), and (−4, 5) that indicate the left branch of the graph.
     • For x > -2: The points (0, −3), (2, 2), and (4, 5) indicate that the right branch of the graph curves and reaches through these points.

3. Identify the vertical asymptote:

   • Since y is undefined at x = -2, we expect a dashed line (vertical asymptote) at x = -2.

From the given options, we can eliminate any options that specify the vertical asymptote at x = -3. Therefore, we only consider options that indicate a vertical dashed line at x = -2.

Analyzing the given options:

• Option 1 mentions a vertical dashed line at x = -2 and shows behavior that matches our expectations (points and asymptote).
• Option 2 and Option 4 mention a vertical asymptote at x = -3, which is not our case.
• Option 3 also mentions the correct vertical dashed line at x = -2 and curving behavior corresponding to the provided points.

Both Option 1 and Option 3 give the vertical asymptote at x = -2, along with consistent behavior regarding the points.

Ultimately, the graph that shows both branches correctly approaching the vertical asymptote at x = -2 and intersects through the given points is represented by Option 3, as it specifically describes the behavior without ambiguity.

Conclusion:

Option 3 is the correct choice that represents the rational function based on the provided values.