Question

the function ( y = f(x) ) is graphed below. what is the average rate of change of the function ( f(x) ) on the interval ( -1 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, \( a=-1 \) and \( b = 4 \).

Step2: Find \( f(-1) \) and \( f(4) \) from the graph

From the graph, when \( x=-1 \), we look at the \( y \)-value. The point at \( x = - 1 \) has a \( y \)-value of \( - 20 \) (since it's on the curve at \( x=-1 \), \( y=-20 \)). When \( x = 4 \), the \( y \)-value is \( 50 \) (from the graph, the point at \( x = 4 \) has \( y = 50 \)).

Step3: Substitute into the formula

Substitute \( a=-1 \), \( b = 4 \), \( f(-1)=-20 \) and \( f(4) = 50 \) into the formula:

$$ \frac{f(4)-f(-1)}{4-(-1)}=\frac{50 - (-20)}{4 + 1}=\frac{50+20}{5}=\frac{70}{5}=14 $$

Answer:

14