The Solution:
Given:
We are required to use the rate of change to determine a function that models the given data.
The rate of change for each step in the data is:
[tex]\begin{gathered} Rate\text{ of change}=\frac{4-1}{1-0}=\frac{3}{1}=3 \\ \\ \text{ Rate of change}=\frac{7-4}{2-1}=\frac{3}{1}=3 \\ \\ \text{Rate of change}=\frac{12-7}{3-2}=\frac{5}{1}=5 \end{gathered}[/tex][tex]\begin{gathered} \text{Rate of change}=\frac{15-12}{4-3}=\frac{3}{1}=3 \\ \\ \text{Rate of change}=\frac{19-15}{5-4}=\frac{4}{1}=4 \\ \\ \text{Rate of change}=\frac{22-19}{6-5}=\frac{3}{1}=3 \end{gathered}[/tex]
So, the average rate of change is:
[tex]Average\text{ rate of change}=\frac{3+3+5+3+4+3}{6}=\frac{21}{6}=3.5[/tex]
Thus, the model for f(x) is:
[tex]f(x)=3.607x+0.6071[/tex]
