To find the interval where the function \( h(x) = -5x + 8 \) has an average rate of change equal to -30, we need to calculate the average rate of change over each of the intervals provided and see if it equals -30.
The average rate of change of a function \( h(x) \) over an interval \([a, b]\) is given by:
\[ \text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a} \]
Let's evaluate the average rate of change for each option:
-
Interval: \( x = -4 \) to \( x = 4 \)
\[ h(-4) = -5(-4) + 8 = 20 + 8 = 28 \] \[ h(4) = -5(4) + 8 = -20 + 8 = -12 \] \[ \text{Average Rate of Change} = \frac{-12 - 28}{4 - (-4)} = \frac{-40}{8} = -5 \]
-
Interval: \( x = 4 \)
This is a single point, so we cannot determine an average rate of change.
-
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 \( x = 4 \)
\[ h(2) = -5(2) + 8 = -10 + 8 = -2 \] \[ h(4) = -5(4) + 8 = -20 + 8 = -12 \] \[ \text{Average Rate of Change} = \frac{-12 - (-2)}{4 - 2} = \frac{-12 + 2}{2} = \frac{-10}{2} = -5 \]
After evaluating all options, none of the given intervals yield an average rate of change of -30. Instead, all calculations return an average rate of change of -5.
Thus, there may be a misunderstanding regarding the average rate of change you seek. If you're looking for a rate of -5, it is consistent with the intervals calculated.
In conclusion, none of the options you've provided yield the specified average rate of -30. Please review the problem or provide additional context if needed.