find the average rate of change of $f(x)=-x^{2}+3x + 3$ from $x = 2$ to $x = 6$. simplify your answer 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, $a = 2$, $b = 6$, and $f(x)=-x^{2}+3x + 3$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=-(2)^{2}+3\times2 + 3=-4 + 6+3=5$.

Step3: Calculate $f(6)$

Substitute $x = 6$ into $f(x)$: $f(6)=-(6)^{2}+3\times6 + 3=-36+18 + 3=-15$.

Step4: Calculate the average rate of change

Substitute $f(2)$ and $f(6)$ into the formula $\frac{f(b)-f(a)}{b - a}$: $\frac{f(6)-f(2)}{6 - 2}=\frac{-15 - 5}{4}=\frac{-20}{4}=-5$.

Answer:

$-5$