Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-5 \leq x \leq -4$?

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, \( a=-5 \) and \( b = - 4 \).

Step2: Find \( f(-5) \) and \( f(-4) \) from the graph

From the graph, we can see that when \( x=-5 \), the \( y \)-value (i.e., \( f(-5) \)) is \(-2\) (since the point is at \((-5, - 2)\)), and when \( x = - 4 \), the \( y \)-value (i.e., \( f(-4) \)) is \(-4\) (since the point is at \((-4, - 4)\)).

Step3: Substitute into the formula

Substitute \( a=-5 \), \( b = - 4 \), \( f(-5)=-2 \) and \( f(-4)=-4 \) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(-4)-f(-5)}{-4-(-5)}=\frac{-4-(-2)}{-4 + 5}=\frac{-4 + 2}{1}=\frac{-2}{1}=-2\).

Answer:

\(-2\)