Question

given the function $h(x) = x^2 + 4x - 4$, determine the average rate of change of the function over the interval $-7 \leq x \leq 0$.

Explanation:

Step1: Recall average rate of change formula

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

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

Here $a=-7$, $b=0$. Calculate $h(-7)$:
$h(-7)=(-7)^2 + 4(-7) - 4 = 49 - 28 - 4 = 17$

Step3: Compute $h(b)$

Calculate $h(0)$:
$h(0)=(0)^2 + 4(0) - 4 = -4$

Step4: Substitute into the formula

$\frac{h(0)-h(-7)}{0-(-7)} = \frac{-4 - 17}{0 + 7} = \frac{-21}{7}$

Step5: Simplify the expression

$\frac{-21}{7} = -3$

Answer:

$-3$