Question

use the graph below to find the average rate of change of the function on the interval -2, 1.

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 \( b = 1 \).

Step2: Determine \( f(-2) \) and \( f(1) \) from the graph

The graph is a V - shaped graph with the vertex at the origin \((0,0)\). For \( x\leq0 \), the graph is a line with slope - 1 (since from \( x = 0,y = 0 \) to \( x=-5,y = 5 \), the slope is \(\frac{5 - 0}{-5-0}=-1\)). So the equation for \( x\leq0 \) is \( y=-x \). When \( x=-2 \), \( f(-2)=-(-2)=2 \).

For \( x\geq0 \), the graph is a line with slope 1 (since from \( x = 0,y = 0 \) to \( x = 5,y = 5 \), the slope is \(\frac{5 - 0}{5-0}=1\)). So the equation for \( x\geq0 \) is \( y = x \). When \( x = 1 \), \( f(1)=1 \).

Step3: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), substitute \( a=-2 \), \( b = 1 \), \( f(-2)=2 \) and \( f(1)=1 \):

$$ \frac{f(1)-f(-2)}{1-(-2)}=\frac{1 - 2}{1 + 2}=\frac{-1}{3}=-\frac{1}{3} $$

Answer:

\(-\frac{1}{3}\)