Question

the graph of a function f is shown below. use the graph of the function to find its average rate of change from x = -4 to x = 2. simplify your answer as much as possible.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-4\) and \(b = 2\).

Step2: Find \(f(-4)\) and \(f(2)\) from the graph

From the graph, when \(x=-4\), \(f(-4)=8\); when \(x = 2\), \(f(2)=-4\).

Step3: Substitute into the formula

Substitute \(a=-4\), \(b = 2\), \(f(-4)=8\), and \(f(2)=-4\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-4 - 8}{2-(-4)}\).

Step4: Simplify the expression

First, simplify the numerator: \(-4-8=-12\). Then simplify the denominator: \(2 + 4=6\). So, \(\frac{-12}{6}=-2\).

Answer:

\(-2\)