find the average rate of change of $f(x)=x^{2}-5x + 1$ from $x = 2$ to $x = 5$. simplify your answer as much…

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, $a = 2$, $b = 5$, and $f(x)=x^{2}-5x + 1$.

Step2: Calculate $f(5)$

Substitute $x = 5$ into $f(x)$: $f(5)=5^{2}-5\times5 + 1=25-25 + 1=1$.

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=2^{2}-5\times2 + 1=4-10 + 1=-5$.

Step4: Calculate the average rate of change

$\frac{f(5)-f(2)}{5 - 2}=\frac{1-(-5)}{3}=\frac{1 + 5}{3}=\frac{6}{3}=2$.

Answer:

$2$