To determine whether the statements regarding the graph are true or not, we should first analyze the given points and what a proportional relationship means.
A proportional relationship is typically described as one in which the ratio of \(y\) to \(x\) is constant. Mathematically, this could be expressed as \(y = kx\) for some constant \(k\).
Let's examine the points provided:
- (0, 2): Here, the ratio \(y/x\) is undefined since we cannot divide by zero when \(x = 0\).
- (2, 3): The ratio is \(3/2 = 1.5\).
- (4, 4): The ratio is \(4/4 = 1\).
- (6, 5): The ratio is \(5/6\).
Each of these ratios is different, indicating that the relationship is not constant.
Now let's analyze 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 is false. While the graph is a line, it does not mean it represents a proportional relationship. The differences between points are not consistent in terms of \(y/x\) ratios.
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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 is also false for the same reason as above. The x-values being multiples of 2 does not guarantee a proportional relationship as defined by a constant ratio.
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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 true, as we have established that the ratios \(y/x\) are not constant.
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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 a bit misleading. The increase in the y-value is not the reason for the absence of proportionality; rather, it's the fact that the ratios between y and x are not consistent.
Thus, the true statement is:
The graph does not show a proportional relationship because each point written as a ratio gives a different value.