Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-5 \leq x \leq 0$?

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 \( b = 0 \).

Step2: Find \( f(-5) \) and \( f(0) \) from the graph

From the graph, when \( x=-5 \), we can see that \( f(-5)=4 \) (looking at the peak around \( x = -5 \) region). When \( x = 0 \), looking at the vertex of the parabola - like part, \( f(0)=-6 \) (since the vertex is at \( (0, - 6) \)).

Step3: Substitute into the formula

Substitute \( a=-5 \), \( b = 0 \), \( f(-5)=4 \) and \( f(0)=-6 \) into the formula:

$$ \frac{f(0)-f(-5)}{0 - (-5)}=\frac{-6 - 4}{0 + 5}=\frac{-10}{5}=-2 $$

Answer:

\(-2\)