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

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

From the graph, when \(x=-6\), \(f(-6)=40\) (by identifying the \(y -\) value of the point on the graph with \(x=-6\)). When \(x = 2\), \(f(2)=0\) (by identifying the \(y -\) value of the point on the graph with \(x = 2\)).

Step3: Substitute values into the formula

Substitute \(a=-6\), \(b = 2\), \(f(a)=40\), and \(f(b)=0\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{0 - 40}{2-(-6)}=\frac{-40}{2 + 6}\).

Step4: Simplify the expression

Simplify \(\frac{-40}{8}=-5\).

Answer:

\(-5\)