Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-6 \leq x \leq -3$?

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, \( a=-6 \) and \( b = - 3 \).

Step2: Find \( f(-6) \) and \( f(-3) \) from the graph

From the graph, when \( x=-6 \), we look at the point on the graph. Let's assume from the graph (by identifying the coordinates) that \( f(-6)=10 \) (since at \( x = - 6 \), the \( y \)-value is 10) and when \( x=-3 \), \( f(-3)=-20 \) (since at \( x=-3 \), the \( y \)-value is - 20).

Step3: Substitute into the formula

Substitute \( a=-6 \), \( b = - 3 \), \( f(-6)=10 \) and \( f(-3)=-20 \) into the formula \(\frac{f(b)-f(a)}{b - a}\).

We get \(\frac{f(-3)-f(-6)}{-3-(-6)}=\frac{-20 - 10}{-3 + 6}=\frac{-30}{3}=-10\).

Answer:

\(-10\)