Question

given the function $h(x) = -x^2 - x + 3$, determine the average rate of change of the function over the interval $-4 \leq x \leq 6$.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $h(x)$ over interval $[a,b]$ is $\frac{h(b)-h(a)}{b-a}$.

Step2: Identify $a$, $b$ and compute $h(a)$

Here $a=-4$, $b=6$. Calculate $h(-4)$:
$h(-4) = -(-4)^2 - (-4) + 3 = -16 + 4 + 3 = -9$

Step3: Compute $h(b)$

Calculate $h(6)$:
$h(6) = -(6)^2 - 6 + 3 = -36 - 6 + 3 = -39$

Step4: Substitute into the formula

$\frac{h(6)-h(-4)}{6-(-4)} = \frac{-39 - (-9)}{6 + 4} = \frac{-30}{10}$

Step5: Simplify the expression

$\frac{-30}{10} = -3$

Answer:

$-3$