Question

find the average rate of change of g(x) on the interval -3 ≤ x ≤ 1

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( g(x) \) on the interval \([a, b]\) is given by \(\frac{g(b) - g(a)}{b - a}\). Here, \( a=-3 \) and \( b = 1 \).

Step2: Find \( g(-3) \) and \( g(1) \) from the graph

From the graph, when \( x=-3 \), \( g(-3)=0 \) (since the graph intersects the \( x \)-axis at \( x = - 3 \)). When \( x = 1 \), we look at the \( y \)-value. From the graph, at \( x = 1 \), \( g(1)=-1 \) (by observing the point on the graph at \( x = 1 \)).

Step3: Substitute into the formula

Substitute \( a=-3 \), \( b = 1 \), \( g(-3)=0 \) and \( g(1)=-1 \) into the average rate of change formula:

$$ \frac{g(1)-g(-3)}{1-(-3)}=\frac{-1 - 0}{1 + 3}=\frac{-1}{4}=-\frac{1}{4} $$

Answer:

\(-\frac{1}{4}\)