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 = 7 to x = 9. 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 \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 7\) and \(b=9\).

Step2: Find \(f(7)\) and \(f(9)\) from the graph

From the graph, when \(x = 7\), \(f(7)=-7\); when \(x = 9\), \(f(9)=-1\).

Step3: Substitute into the formula

Substitute \(a = 7\), \(b = 9\), \(f(7)=-7\), and \(f(9)=-1\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{-1-(-7)}{9 - 7}=\frac{-1 + 7}{2}\).

Step4: Simplify the expression

\(\frac{-1 + 7}{2}=\frac{6}{2}=3\).

Answer:

\(3\)