given the function $f(x)=-x^{2}+3x + 5$, determine the average rate of change of the function over the…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$, $b = 5$, and $f(x)=-x^{2}+3x + 5$.

Step2: Calculate $f(5)$

Substitute $x = 5$ into $f(x)$: $f(5)=-(5)^{2}+3\times5 + 5=-25 + 15+5=-5$.

Step3: Calculate $f(1)$

Substitute $x = 1$ into $f(x)$: $f(1)=-(1)^{2}+3\times1 + 5=-1 + 3+5=7$.

Step4: Calculate the average rate of change

$\frac{f(5)-f(1)}{5 - 1}=\frac{-5 - 7}{4}=\frac{-12}{4}=-3$.

Answer:

$-3$