Question

estimate the average rate of change from $x = - 1$ to $x = 2$

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-1\) and \(b = 2\).

Step2: Estimate \(f(-1)\) and \(f(2)\) from the graph

From the graph, when \(x=-1\), \(f(-1)\approx3\) (by looking at the \(y -\) value of the open - circle point). When \(x = 2\), \(f(2)=5\) (by looking at the \(y -\) value of the closed - circle point).

Step3: Substitute into the formula

Substitute \(a=-1\), \(b = 2\), \(f(a)\approx3\), and \(f(b)=5\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{5 - 3}{2-(-1)}=\frac{2}{3}\approx0.67\).

Answer:

The average rate of change is approximately \(\frac{2}{3}\) (or about \(0.67\)).