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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


next up previous
Next: Quadratic Functions Up: Linear Functions-Planes Previous: Using any three non-collinear
Vector Calculus
1/8/1998