To find the interval where the function \(h(x) = -5x + 8\) has an average rate of change equal to \(-30\), we can use the formula for the average rate of change of a function over the interval \([a, b]\):
\[ \text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a} \]
Let's evaluate each interval you provided:
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Interval \(x = -4\) to \(x = 4\):
- \(h(-4) = -5(-4) + 8 = 20 + 8 = 28\)
- \(h(4) = -5(4) + 8 = -20 + 8 = -12\)
\[ \text{Average Rate of Change} = \frac{-12 - 28}{4 - (-4)} = \frac{-40}{8} = -5 \]
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At \(x = 4\):
- This is just a point evaluation, not an interval. The average rate of change cannot be calculated for a single point.
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Interval \(x = 0\) to \(x = 2\):
- \(h(0) = -5(0) + 8 = 8\)
- \(h(2) = -5(2) + 8 = -10 + 8 = -2\)
\[ \text{Average Rate of Change} = \frac{-2 - 8}{2 - 0} = \frac{-10}{2} = -5 \]
-
Interval \(x = 2\) to \(x = 4\):
- \(h(2) = -5(2) + 8 = -2\) (calculated previously)
- \(h(4) = -12\) (calculated previously)
\[ \text{Average Rate of Change} = \frac{-12 - (-2)}{4 - 2} = \frac{-12 + 2}{2} = \frac{-10}{2} = -5 \]
Summary of Results:
- Average rate of change from \([-4, 4]\) is \(-5\)
- Average rate of change from \([0, 2]\) is \(-5\)
- Average rate of change from \([2, 4]\) is \(-5\)
None of the given intervals yield an average rate of change of \(-30\). All calculated averages are equal to \(-5\), indicating that \(h(x)\) has a steady rate of change over the intervals, consistent with the linear nature of the function.
Thus, none of the intervals provide an average rate of change that is equal to or closer to -30.