given the function $f(x)=-x^{2}+5x + 14$, 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, $a = 0$, $b = 9$, and $f(x)=-x^{2}+5x + 14$.

Step2: Calculate $f(9)$

Substitute $x = 9$ into $f(x)$: $f(9)=-(9)^{2}+5\times9 + 14=-81 + 45+14=-22$.

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-(0)^{2}+5\times0 + 14=14$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(9)-f(0)}{9 - 0}=\frac{-22 - 14}{9}=\frac{-36}{9}=-4$.

Answer:

$-4$