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 = -5 to x = -3. 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, \(a=-5\) and \(b = - 3\).

Step2: Find \(f(-5)\) and \(f(-3)\) from the graph

From the graph, when \(x=-5\), \(y=-6\) (so \(f(-5)=-6\)), and when \(x=-3\), \(y = 1\) (so \(f(-3)=1\)).

Step3: Substitute into the formula

Substitute \(a=-5\), \(b=-3\), \(f(a)=-6\), and \(f(b) = 1\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{1-(-6)}{-3-(-5)}=\frac{1 + 6}{-3 + 5}=\frac{7}{2}\).

Answer:

\(\frac{7}{2}\)