Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-1 \leq x \leq 4$?

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, \( a=-1 \) and \( b = 4 \).

Step2: Find \( f(-1) \) from the graph

Looking at the graph, when \( x=-1 \), we need to find the corresponding \( y \)-value. From the graph, at \( x=-1 \), the point is on the left - hand curve. By observing the graph (the grid and the plotted points), we can see that when \( x=-1 \), \( f(-1)=0 \) (since it's on the x - axis).

Step3: Find \( f(4) \) from the graph

When \( x = 4 \), looking at the graph, the function passes through the x - axis at \( x = 4 \), so \( f(4)=0 \).

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), substitute \( a=-1 \), \( b = 4 \), \( f(-1)=0 \) and \( f(4)=0 \) into it. We get \(\frac{f(4)-f(-1)}{4-(-1)}=\frac{0 - 0}{4 + 1}=\frac{0}{5}=0\).

Answer:

\( 0 \)