question given the function g(x)= -x² - 2x + 6, determine the average rate of change of the function over…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$, where $a=-7$ and $b = 3$.

Step2: Calculate $g(a)$

Substitute $x=-7$ into $g(x)=-x^{2}-2x + 6$. $g(-7)=-(-7)^{2}-2\times(-7)+6=-49 + 14+6=-29$.

Step3: Calculate $g(b)$

Substitute $x = 3$ into $g(x)=-x^{2}-2x + 6$. $g(3)=-3^{2}-2\times3+6=-9-6 + 6=-9$.

Step4: Calculate the average rate of change

$\frac{g(3)-g(-7)}{3-(-7)}=\frac{-9-(-29)}{3 + 7}=\frac{-9 + 29}{10}=\frac{20}{10}=2$.

Answer:

$2$