Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-3 \leq x \leq 3$? (graph is shown with x-axis from -10 to 10 and y-axis from -20 to 20, with the functions graph plotted)

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, \( a=-3 \) and \( b = 3 \).

Step2: Find \( f(-3) \) from the graph

Looking at the graph, when \( x=-3 \), we need to determine the \( y \)-value. From the graph's left - hand part (the curve on the left), we can see that when \( x=-3 \), the \( y \)-value ( \( f(-3) \)) is \(-8\) (by looking at the coordinates of the point on the graph at \( x = - 3 \)).

Step3: Find \( f(3) \) from the graph

Looking at the graph, when \( x = 3 \), we determine the \( y \)-value. From the graph, when \( x=3 \), the \( y \)-value ( \( f(3) \)) is \(-12\) (by looking at the coordinates of the point on the graph at \( x = 3 \)).

Step4: Calculate the average rate of change

Substitute \( a=-3 \), \( b = 3 \), \( f(-3)=-8 \) and \( f(3)=-12 \) into the formula \(\frac{f(b)-f(a)}{b - a}\).
\[

$$\begin{align*} \text{Average rate of change}&=\frac{f(3)-f(-3)}{3-(-3)}\\ &=\frac{-12-(-8)}{3 + 3}\\ &=\frac{-12 + 8}{6}\\ &=\frac{-4}{6}\\ &=-\frac{2}{3} \end{align*}$$

\]

Answer:

\(-\frac{2}{3}\)