Question

use the graph below to find the average rate of change of the function on the interval 1,8.

Explanation:

Step1: Identify the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\) and \(b=8\).

Step2: Find \(f(1)\) and \(f(8)\) from the graph

From the graph, when \(x = 1\), \(y=f(1)=0\) (since the vertex of the V - shaped graph is at \((5,0)\) and the left - hand side of the graph is symmetric). When \(x = 8\), we can calculate the slope of the right - hand side of the graph. The right - hand side of the graph is a line. Let's assume two points on the right - hand side: \((5,0)\) and \((20,20)\). The slope \(m=\frac{20 - 0}{20 - 5}=\frac{4}{3}\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(5,0)\), we get \(y=\frac{4}{3}(x - 5)\). When \(x = 8\), \(y=\frac{4}{3}(8 - 5)=4\), so \(f(8)=4\).

Step3: Calculate the average rate of change

Substitute \(a = 1\), \(b = 8\), \(f(1)=0\) and \(f(8)=4\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We have \(\frac{4-0}{8 - 1}=\frac{4}{7}\).

Answer:

\(\frac{4}{7}\)