find the average rate of change of $g(x)=-2x^{2}-2x$ from $x = 1$ to $x = 6$. simplify your answer as much…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ from $x = a$ to $x = b$ is $\frac{g(b)-g(a)}{b - a}$. Here, $a = 1$, $b = 6$, and $g(x)=-2x^{2}-2x$.

Step2: Calculate $g(6)$

Substitute $x = 6$ into $g(x)$: $g(6)=-2\times6^{2}-2\times6=-2\times36 - 12=-72-12=-84$.

Step3: Calculate $g(1)$

Substitute $x = 1$ into $g(x)$: $g(1)=-2\times1^{2}-2\times1=-2 - 2=-4$.

Step4: Calculate the average rate of change

$\frac{g(6)-g(1)}{6 - 1}=\frac{-84-(-4)}{5}=\frac{-84 + 4}{5}=\frac{-80}{5}=-16$.

Answer:

$-16$