Question

the function ( y = f(x) ) is graphed below. what is the average rate of change of the function ( f(x) ) on the interval ( 2 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 \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 2\) and \(b=6\).

Step2: Find \(f(2)\) and \(f(6)\) from the graph

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

Step3: Substitute into the formula

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

Step4: Simplify the expression

\(\frac{-6 - 2}{6 - 2}=\frac{-8}{4}=-2\)

Answer:

\(-2\)