Question

find the average rate of change of $f(x)=x^2 -4x + 1$ from $x = 3$ to $x = 7$. simplify your answer as much as possible.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 3 \), \( b = 7 \), and \( f(x)=x^{2}-4x + 1 \).

Step2: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=(3)^{2}-4(3)+1=9 - 12 + 1=-2 \)

Step3: Calculate \( f(7) \)

Substitute \( x = 7 \) into \( f(x) \):
\( f(7)=(7)^{2}-4(7)+1=49 - 28 + 1=22 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(7)-f(3)}{7 - 3} \), substitute \( f(7) = 22 \) and \( f(3)=-2 \):
\( \frac{22-(-2)}{7 - 3}=\frac{22 + 2}{4}=\frac{24}{4}=6 \)

Answer:

\( 6 \)