To determine the average rate of change of the function [tex]\( h(x) = -x^2 - 5x + 9 \)[/tex] over the interval [tex]\( -8 \leq x \leq 2 \)[/tex], follow these steps:
1. Evaluate the function at the endpoints of the interval: We need to find the values of [tex]\( h(x) \)[/tex] at [tex]\( x = -8 \)[/tex] and [tex]\( x = 2 \)[/tex].
2. Calculate [tex]\( h(x) \)[/tex] at [tex]\( x = -8 \)[/tex]:
[tex]\[
h(-8) = -(-8)^2 - 5(-8) + 9
= -64 + 40 + 9
= -15
\][/tex]
3. Calculate [tex]\( h(x) \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[
h(2) = -(2)^2 - 5(2) + 9
= -4 - 10 + 9
= -5
\][/tex]
4. Apply the average rate of change formula: The formula for the average rate of change of [tex]\( h(x) \)[/tex] from [tex]\( x = a \)[/tex] to [tex]\( x = b \)[/tex] is:
[tex]\[
\frac{h(b) - h(a)}{b - a}
\][/tex]
In this case, [tex]\( a = -8 \)[/tex] and [tex]\( b = 2 \)[/tex]. So, we plug in the values:
[tex]\[
\frac{h(2) - h(-8)}{2 - (-8)} = \frac{-5 - (-15)}{2 - (-8)} = \frac{-5 + 15}{2 + 8} = \frac{10}{10} = 1
\][/tex]
Therefore, the average rate of change of the function [tex]\( h(x) = -x^2 - 5x + 9 \)[/tex] over the interval [tex]\( -8 \leq x \leq 2 \)[/tex] is [tex]\( 1 \)[/tex].
Here are the results:
- [tex]\( h(-8) = -15 \)[/tex]
- [tex]\( h(2) = -5 \)[/tex]
- Average rate of change: [tex]\( 1 \)[/tex]
