Question

the function ( y = f(x) ) is graphed below. what is the average rate of change of the function ( f(x) ) on the interval ( -6 leq x leq -2 )?

Explanation:

Step1: Recall the formula for average rate of change

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, \(a=-6\) and \(b = - 2\).

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

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

Step3: Substitute into the formula

Substitute \(a=-6\), \(b=-2\), \(f(a)=-4\), and \(f(b)=-6\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-6-(-4)}{-2-(-6)}=\frac{-6 + 4}{-2 + 6}=\frac{-2}{4}=-\frac{1}{2}\)

Answer:

\(-\frac{1}{2}\)