question given the function g(x)= -x² - 5x + 11, 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 $\frac{g(b)-g(a)}{b - a}$. Here, $a=-8$ and $b = 4$.

Step2: Calculate $g(a)$

Substitute $x=-8$ into $g(x)=-x^{2}-5x + 11$. $g(-8)=-(-8)^{2}-5\times(-8)+11=-64 + 40+11=-13$.

Step3: Calculate $g(b)$

Substitute $x = 4$ into $g(x)=-x^{2}-5x + 11$. $g(4)=-4^{2}-5\times4+11=-16-20 + 11=-25$.

Step4: Calculate the average rate of change

$\frac{g(4)-g(-8)}{4-(-8)}=\frac{-25-(-13)}{4 + 8}=\frac{-25 + 13}{12}=\frac{-12}{12}=-1$.

Answer:

$-1$