Linear Functions and Tables

Since we know that the slope of a linear function in the x and y directions is constant, it is relatively easy to check whether a function given by a table of values is linear. Suppose the following table represents a function of x and y.

X

-2

-1

0

1

2

Y

-2

-1

0

1

2

3	1	-1	-3	-5
4	2	0	-2	-4
5	3	1	-1	-3
6	4	2	0	-2
7	5	3	1	-1

To determine whether or not this represents a linear function, go column by column to check whether $\Delta z/\Delta y$ is constant and then go row by row to check whether $\Delta z/\Delta x$ is constant. In this case, any pair of entries in any column produces $\Delta z/\Delta y = 1$ and any pair of entries in any row produces $\Delta z/\Delta x = -2$. Thus, this function is indeed linear.

However, the function shown in the table below is not a linear function, since $\Delta z/\Delta y$ is not the same for any two entries in a column.

X

-4

-2

0

2

4

Y

-4

-2

0

2

4

5	7	9	11	13
6	8	10	12	14
8	10	12	14	16
9	11	13	15	17
10	12	14	16	18