let $f(x)=-\frac{1}{2}(x + 2)^2+5$. what is the average rate of change for the quadratic function from…

Answer:

$-\frac{3}{2}$

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-3$, $b = 1$.

Step2: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)=-\frac{1}{2}(x + 2)^2+5$. $f(-3)=-\frac{1}{2}(-3 + 2)^2+5=-\frac{1}{2}(-1)^2+5=-\frac{1}{2}+5=\frac{-1 + 10}{2}=\frac{9}{2}$.

Step3: Calculate $f(1)$

Substitute $x = 1$ into $f(x)=-\frac{1}{2}(x + 2)^2+5$. $f(1)=-\frac{1}{2}(1 + 2)^2+5=-\frac{1}{2}(3)^2+5=-\frac{9}{2}+5=\frac{-9 + 10}{2}=\frac{1}{2}$.

Step4: Calculate average rate of change

$\frac{f(1)-f(-3)}{1-(-3)}=\frac{\frac{1}{2}-\frac{9}{2}}{1 + 3}=\frac{\frac{1 - 9}{2}}{4}=\frac{-\frac{8}{2}}{4}=\frac{-4}{4}=-\frac{3}{2}$.