Question

the graph shows the distance in miles of a runner over x hours. what is the average rate of speed over the interval 9,11?

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 9\), \(b=11\). We need to find \(f(9)\) and \(f(11)\) from the graph.

Step2: Identify the values of \(f(9)\) and \(f(11)\)

From the graph, when \(x = 9\), assume \(f(9)\approx7\) (by interpolation, since the points \((8,6)\) and \((11,11)\) are given. The \(x\) - values have a difference of \(11-8 = 3\) and \(y\) - values have a difference of \(11 - 6=5\). For \(x\) from \(8\) to \(9\) (a change of \(\Delta x=1\)), \(\Delta y=\frac{5}{3}\approx1.67\), so \(f(9)=6 + 1.67\approx7\)). When \(x = 11\), \(f(11)=11\)

Step3: Apply the formula

Substitute into the average - rate - of - change formula: \(\frac{f(11)-f(9)}{11 - 9}=\frac{11 - 7}{2}\)

$$=\frac{4}{2}=2$$

Answer:

\(2\) miles per hour