Question

the function $y = f(x)$ is graphed below. what is the average rate of change of the function $f(x)$ on the interval $-7 \leq x \leq -5$?

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, \( a = -7 \) and \( b = -5 \).

Step2: Find \( f(-7) \) and \( f(-5) \) from the graph

From the graph, when \( x = -7 \), the corresponding \( y \)-value (i.e., \( f(-7) \)) is \( 0 \) (since the point is on the x - axis at \( x=-7 \)). When \( x = -5 \), the corresponding \( y \)-value (i.e., \( f(-5) \)) is \( 4 \) (by looking at the graph, the point at \( x = -5 \) has a \( y \)-coordinate of \( 4 \)).

Step3: Substitute into the formula

Substitute \( a=-7 \), \( b = -5 \), \( f(-7)=0 \) and \( f(-5) = 4 \) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(-5)-f(-7)}{-5-(-7)}=\frac{4 - 0}{-5 + 7}=\frac{4}{2}=2\).

Answer:

\( 2 \)