To find the rate of change from the given table, we can use the formula for the rate of change, which is the change in \(Y\) divided by the change in \(X\).
From the table:
- When \(X = 1\), \(Y = 8\)
- When \(X = 2\), \(Y = 4\)
- When \(X = 3\), \(Y = 0\)
- When \(X = 4\), \(Y = -4\)
Let's calculate the rate of change between the first pair of points (from \(X = 1\) to \(X = 2\)):
\[ \text{Rate of Change} = \frac{Y_2 - Y_1}{X_2 - X_1} = \frac{4 - 8}{2 - 1} = \frac{-4}{1} = -4 \]
Now let's check another segment for consistency—between \(X = 2\) and \(X = 3\):
\[ \text{Rate of Change} = \frac{Y_3 - Y_2}{X_3 - X_2} = \frac{0 - 4}{3 - 2} = \frac{-4}{1} = -4 \]
And between \(X = 3\) and \(X = 4\):
\[ \text{Rate of Change} = \frac{Y_4 - Y_3}{X_4 - X_3} = \frac{-4 - 0}{4 - 3} = \frac{-4}{1} = -4 \]
In all segments, the rate of change is \(-4\).
Thus, the rate of change for the entire set of data is \(-4\).
The answer is C. -4.
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