Writing a Linear Function's Rule from Two Pairs

A line has only two personality traits: how steep it is and where it starts. Those are m and b in f(x) = mx + b. Give two input-output pairs and you've pinned down both, because a straight line is the unique path between two points — there's no curve, no bend, nowhere else it could go. Once you know the rate and one point, the whole function is locked in, for every x, forever.

Start with the rate. Given f(x1) = y1 and f(x2) = y2, the rate of change is m = (y2 - y1) / (x2 - x1), the same slope formula from coordinate geometry, just dressed in function notation. Once you have m, pick either pair — it doesn't matter which — and plug into f(x) = mx + b. Solve for b by substituting the known x and f(x). That b is the output when x = 0, the function's starting value.

This skill is the function-notation twin of writing an equation of a line from two points, and it feeds directly into systems of equations, since two linear functions written this way can be set equal to find where they cross. It also sets up transformations: once you have f(x) = mx + b, shifting, scaling, or comparing it to another linear function becomes a matter of comparing m's and b's directly.