To determine whether the graph represents a proportional relationship, we need to evaluate the statements and the nature of the points provided:
- The points given are (0, 2), (2, 3), (4, 4), and (6, 5).
- For a relationship to be proportional, it needs to pass through the origin (0,0) and the ratios of y to x (y/x) must be constant for any point on the line.
Now let's analyze the points:
- The point (0, 2) indicates that when x = 0, y = 2, which means the line does not pass through the origin, thus failing the first condition necessary for a proportional relationship.
Next, we can check the ratios of y to x for the other points:
- For (2, 3): \( \frac{y}{x} = \frac{3}{2} = 1.5 \)
- For (4, 4): \( \frac{y}{x} = \frac{4}{4} = 1 \)
- For (6, 5): \( \frac{y}{x} = \frac{5}{6} \approx 0.833 \)
Since the ratios are not constant, the relationship is not proportional.
Now reviewing the statements:
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The graph shows a proportional relationship because it is a line, and the difference between each point is the same.
- This statement is incorrect because a line alone does not guarantee proportionality, especially since the line does not pass through the origin.
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The graph shows a proportional relationship because it is a line, and each x-value is a multiple of 2.
- This statement is incorrect for the same reason as above.
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The graph does not show a proportional relationship because each point written as a ratio gives a different value.
- This statement is correct; the varying ratios confirm that the relationship is not proportional.
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The graph does not show a proportional relationship because a line that increases by 1 in the y-value cannot have a constant of proportionality.
- This statement is misleading; the point is not about the increase rather that it doesn’t pass through the origin.
The true statement about the graph is: The graph does not show a proportional relationship because each point written as a ratio gives a different value.
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