How to Identify a Linear Function from a Table

Linear tables have a constant rate of change

A table represents a linear function when equal changes in $x$ produce equal changes in $y$. Between any two rows, calculate

$$m=\frac{\Delta y}{\Delta x} =\frac{y_2-y_1}{x_2-x_1}.$$

For one line, every pair of consecutive rows must give the same slope $m$. This is the tabular version of a straight graph and the equation $y=mx+b$.

Constant first differences are a shortcut, not the full definition

When $x$ increases by the same amount in every row, the denominators $\Delta x$ are equal. You can then compare only the first differences in $y$.

This table has first differences $0.5,0.5,0.5$, so its slope is $0.5$.

If the $x$-spacing is uneven, equal $y$-differences are not enough. Compare each quotient $\Delta y/\Delta x$ instead. A rise of 4 over a run of 2 has the same slope as a rise of 6 over a run of 3.

Recover the equation from the table

After finding $m$, substitute any row into $y=mx+b$:

$$b=y-mx.$$

Then verify the equation against every row. If $b=0$, the line passes through the origin and the relationship is proportional. If $b\ne0$, it is still linear but not proportional.

Common nonlinear patterns

A changing first difference signals that the rate is changing. Quadratic tables with equally spaced inputs have changing first differences but constant second differences. Exponential tables often have a constant ratio between successive outputs. Reciprocal tables such as $y=1/x$ have neither a constant first difference nor a constant slope.

Recognizing those patterns can explain why a table is nonlinear, but the decisive linearity test remains constant $\Delta y/\Delta x$.

Avoid three frequent mistakes

First, do not inspect only one interval; two points always determine a line, so at least three points are needed to test whether the rate stays constant. Second, keep the rows in their $x$ order so consecutive differences are meaningful. Third, do not confuse “function” with “linear function.” A table can assign exactly one output to every input and still be nonlinear.

A compact verification routine

Order the rows by $x$, calculate every consecutive $\Delta x$ and $\Delta y$, reduce each slope, and reject the table if one slope differs. If all slopes agree, calculate $b$ from one row and substitute the remaining rows into $y=mx+b$. This both identifies the linear table and supplies its equation.