To determine if the data in the table represents a linear or nonlinear function, we need to check if there is a constant rate of change (slope) between the pairs of (x, y) values.
Let's calculate the changes in y divided by the changes in x:
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From \(x = -4\) to \(x = -1\): \[ \Delta y = 2.5 - 4 = -1.5 \quad \Delta x = -1 - (-4) = 3 \quad \text{slope} = \frac{-1.5}{3} = -0.5 \]
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From \(x = -1\) to \(x = 0\): \[ \Delta y = 2 - 2.5 = -0.5 \quad \Delta x = 0 - (-1) = 1 \quad \text{slope} = \frac{-0.5}{1} = -0.5 \]
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From \(x = 0\) to \(x = 4\): \[ \Delta y = 0 - 2 = -2 \quad \Delta x = 4 - 0 = 4 \quad \text{slope} = \frac{-2}{4} = -0.5 \]
In this case:
- From \(x = -4\) to \(x = -1\), the slope is \(-0.5\).
- From \(x = -1\) to \(x = 0\), the slope is \(-0.5\).
- From \(x = 0\) to \(x = 4\), the slope is also \(-0.5\).
Since the slope (rate of change) is constant for all intervals, the data represents a linear function.
Based on this assessment, the correct statement is:
The table represents a linear function because the graph shows a constant rate of change.