what is the average rate of change of the function $g(x)=-x^{2}-2x + 1$ on the interval from $x = 2$ to $x =…

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 $\frac{g(b)-g(a)}{b - a}$. Here, $a = 2$, $b = 6$, and $g(x)=-x^{2}-2x + 1$.

Step2: Calculate $g(2)$

Substitute $x = 2$ into $g(x)$: $g(2)=-(2)^{2}-2\times2 + 1=-4-4 + 1=-7$.

Step3: Calculate $g(6)$

Substitute $x = 6$ into $g(x)$: $g(6)=-(6)^{2}-2\times6+1=-36-12 + 1=-47$.

Step4: Calculate the average rate of change

$\frac{g(6)-g(2)}{6 - 2}=\frac{-47-(-7)}{4}=\frac{-47 + 7}{4}=\frac{-40}{4}=-10$.

Answer:

$-10$