find the average rate of change of f(x)=-x^2 + 3x + 2 from x = 2 to x = 4. simplify your answer as much as…

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, $a = 2$, $b = 4$, and $f(x)=-x^{2}+3x + 2$.

Step2: Calculate $f(4)$

Substitute $x = 4$ into $f(x)$: $f(4)=-(4)^{2}+3\times4 + 2=-16 + 12+2=-2$.

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=-(2)^{2}+3\times2 + 2=-4 + 6+2=4$.

Step4: Calculate the average rate of change

$\frac{f(4)-f(2)}{4 - 2}=\frac{-2 - 4}{2}=\frac{-6}{2}=-3$.

Answer:

$-3$