Question

on the graph above, estimate to one decimal the average rate of change from $x = 1$ to $x = 5$

Explanation:

Step1: Recall the average rate of change formula

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, \(a = 1\) and \(b=5\).

Step2: Estimate \(f(1)\) and \(f(5)\) from the graph

From the graph, when \(x = 1\), \(f(1)\approx3.5\). When \(x = 5\), \(f(5)\approx1.8\).

Step3: Substitute into the formula

Substitute \(a = 1\), \(b = 5\), \(f(1)\approx3.5\), and \(f(5)\approx1.8\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{1.8-3.5}{5 - 1}=\frac{- 1.7}{4}=-0.425\approx - 0.4\)

Answer:

\(-0.4\)