Question

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. which best describes the data set? 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 \total houses built\ column does not increase at a constant additive rate. 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. it is nonlinear because the \months passed\ column does not increase at a constant additive rate. a linear function has a constant rate of change. determine the rate of change between consecutive points on the table. if the rate of change is constant, the data in the table is linear. if the rate of change is not constant, the data in the table is nonlinear.

Explanation:

Step1: Recall the definition of linear function

A linear function has a constant rate of change. To check the rate of change between two - variable data (months passed and total houses built), we use the formula for rate of change $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$, where $y$ is the total number of houses built and $x$ is the number of months passed.

Step2: Calculate the rate of change between the first two data - points

Let $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(3,33)$. The rate of change $r_1=\frac{33 - 0}{3 - 0}=\frac{33}{3}=11$.

Step3: Calculate the rate of change between the second and third data - points

Let $(x_1,y_1)=(3,33)$ and $(x_2,y_2)=(4,46)$. The rate of change $r_2=\frac{46 - 33}{4 - 3}=\frac{13}{1}=13$.

Step4: Analyze the rates of change

Since $r_1 = 11$ and $r_2=13$, the rate of change is not constant. A non - constant rate of change indicates a non - linear function.

Answer:

It is nonlinear because the "Total Houses Built" compared to the "Months Passed" does not show a constant rate of change.