Question

Does this table represent a linear function or a non-linear function?

Answer 1

Solution:

Concept:

To figure out if the value on the table is a linear or non-linear function, we will calculate their slopes individually to give the same value

To determine whether the function is linear or nonlinear, see whether it has a constant rate of change. Pick the points in any two rows of the table and calculate the rate of change between them. The first two rows are a good place to start. Now pick any other two rows and calculate the rate of change between them.

The coordinates of the table are given below as

$\begin{gathered} (-1,2)\Rightarrow x_1=-1,y_1=2 \\ (0,4)\Rightarrow x_2=0,y_2=4 \\ (1,6)\Rightarrow x_3=1,y_3=6 \\ (2,8)\Rightarrow x_4=2,y_4=8 \end{gathered}$

Step 1:

Calculate the slope using the formula below

$\text{slope}=m=\frac{y_2-y_1}{x_2-x_1}$

By substituting the values, we will have

$\begin{gathered} \text{slope}=m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{4-2}{0-(-1)} \\ m=\frac{2}{1} \\ m=2 \end{gathered}$

Step 2:

Calculate the slope using the formula below

$m=\frac{y_3-y_2}{x_3-x_2}$

By substituting the values, we will have

$\begin{gathered} m=\frac{y_3-y_2}{x_3-x_2} \\ m=\frac{6-4}{1-0} \\ m=\frac{2}{1} \\ m=2 \end{gathered}$

Step 3:

Calculate the slope using the formula below

$m=\frac{y_4-y_3}{x_3-x_2}$

By substituting the values, we will have

$\begin{gathered} m=\frac{y_4-y_3}{x_3-x_2} \\ m=\frac{8-6}{2-1} \\ m=\frac{2}{1} \\ m=2 \end{gathered}$

From the calculations above, we can see that the slopes remain constant using different pairs of coordinates on the table of values.

Using a graphing calculator, we will have the graph of the table to be

To determine the function of the line on the table of values, we will use the formula below

$m=\frac{y-y_1}{x-x_1}$

By substituting the values, we will have

$\begin{gathered} m=\frac{y-y_1}{x-x_1} \\ \frac{2}{1}=\frac{y-2}{x-(-1)} \\ \frac{2}{1}=\frac{y-2}{x+1} \\ 1(y-2)=2(x+1) \\ y-2=2x+2 \\ y=2x+2+2 \\ y=2x+4 \end{gathered}$

Hence,

The table represents a LINEAR function

The first OPTION is the right answer