Question

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

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 3\) and \(b=6\).

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

From the graph, when \(x = 3\), \(f(3)=2\); when \(x = 6\), \(f(6)=- 4\).

Step3: Substitute values into the formula

Substitute \(a = 3\), \(b = 6\), \(f(3)=2\), and \(f(6)=-4\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(6)-f(3)}{6 - 3}=\frac{-4 - 2}{6-3}\).

Step4: Simplify the expression

\(\frac{-4 - 2}{6 - 3}=\frac{-6}{3}=-2\)

Answer:

\(-2\)