Question

introduction to linear functions - analyzing a table to determine if the function is linear or nonlinear. the function in the table shows the relationship between the total number of houses built in an area and the number of months that passed. months passed: 0, 3, 4, 8. total houses built: 0, 33, 46, 108. which best describes the data set? it is nonlinear because the \total houses built\ column does not increase at a constant additive rate. it is nonlinear because the \months passed\ column does not increase at a constant additive rate. it is linear because the increase in the \total houses built\ compared to the increase in the \months passed\ shows a constant rate of change. it is nonlinear because the increase in the \total houses built\ compared to the increase in the \months passed\ does not show a constant rate of change.

Explanation:

Step1: Recall linear - function property

A linear function has a constant rate of change. To check if the relationship between months passed and total houses built is linear, we calculate the rate of change.

Step2: Calculate rate of change between first two points

For months passed \(x_1 = 0,x_2 = 3\) and total houses built \(y_1=0,y_2 = 33\). The rate of change \(m_1=\frac{y_2 - y_1}{x_2 - x_1}=\frac{33 - 0}{3 - 0}=11\).

Step3: Calculate rate of change between second and third points

For \(x_2 = 3,x_3 = 4\) and \(y_2 = 33,y_3 = 46\). The rate of change \(m_2=\frac{y_3 - y_2}{x_3 - x_2}=\frac{46 - 33}{4 - 3}=13\).

Step4: Analyze the rates of change

Since \(m_1
eq m_2\), the total number of houses built does not increase at a constant rate with respect to the number of months passed.

Answer:

It is nonlinear because the "Total Houses Built" column does not increase at a constant additive rate.