Table, Equation, and Graph of a Linear Function

A linear function has one fixed rule: every time the input goes up by some fixed amount, the output goes up by a matching fixed amount. A table shows this as rows of numbers, an equation shows it as f(x) = mx + b, and a graph shows it as a straight line. They are not three different things to memorize — they are three readouts of the same rate and the same starting value. Once you see the rate in one form, you can produce the other two.

In a table, you find the slope by picking two rows and dividing the change in output by the change in input: m = (y2 - y1) / (x2 - x1). This only works cleanly when you divide by the actual input gap — if the inputs jump by 2 instead of 1, you must divide by 2, not treat the output jump alone as the slope. Once you have m, plug one known pair into y = mx + b and solve for b, or build the equation directly from point-slope form. From a graph, the same idea works: pick two clearly plotted points, read the rise and run between them, and use that ratio as the slope.

This topic is the hinge between seeing a function and manipulating one. Everything after it — systems of equations, inequalities, modeling word problems — assumes you can slide freely between a table someone handed you, an equation you can solve, and a picture you can read.