Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $2 \leq x \leq 6$?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 2 \) and \( b = 6 \).

Step2: Find \( f(2) \) and \( f(6) \) from the graph

From the graph, when \( x = 2 \), we look at the point on the graph. The \( y \)-coordinate (value of \( f(2) \)) is \(-20\) (since the point is at \((2, -20)\)). When \( x = 6 \), the \( y \)-coordinate (value of \( f(6) \)) is \( 0 \) (since the graph crosses the \( x \)-axis at \( x = 6 \), so \( f(6)=0 \)).

Step3: Calculate the average rate of change

Substitute \( a = 2 \), \( b = 6 \), \( f(2)=-20 \), and \( f(6)=0 \) into the formula:

$$ \frac{f(6) - f(2)}{6 - 2}=\frac{0 - (-20)}{4}=\frac{20}{4} = 5 $$

Answer:

The average rate of change is \( 5 \).