find the average rate of change of $f(x)=x^{2}-5x + 1$ from $x = 3$ to $x = 7$. 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, $a = 3$, $b = 7$, and $f(x)=x^{2}-5x + 1$.

Step2: Calculate $f(7)$

Substitute $x = 7$ into $f(x)$: $f(7)=7^{2}-5\times7 + 1=49-35 + 1=15$.

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=3^{2}-5\times3 + 1=9-15 + 1=-5$.

Step4: Calculate the average rate of change

$\frac{f(7)-f(3)}{7 - 3}=\frac{15-(-5)}{4}=\frac{15 + 5}{4}=\frac{20}{4}=5$.

Answer:

$5$