given the function $f(x)=x^{2}-9x + 16$, determine the average rate of change of the function over the…

given the function $f(x)=x^{2}-9x + 16$, determine the average rate of change of the function over the interval $-1leq xleq9$.

given the function $f(x)=x^{2}-9x + 16$, determine the average rate of change of the function over the interval $-1leq xleq9$.

Answer

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, $a=-1$, $b = 9$, and $f(x)=x^{2}-9x + 16$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$: $f(-1)=(-1)^{2}-9\times(-1)+16=1 + 9+16=26$.

Step3: Calculate $f(9)$

Substitute $x = 9$ into $f(x)$: $f(9)=9^{2}-9\times9+16=81-81 + 16=16$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(9)-f(-1)}{9-(-1)}=\frac{16 - 26}{9 + 1}=\frac{-10}{10}=-1$.

Answer:

$-1$