Building Functions for Proportional and Linear Relations (F 503)

Proportional means the ratio of output to input never changes. If y is proportional to x, then y/x = k for every pair, so y = kx. Plug in one matched pair, solve for k, and you have the whole rule. Because y = k(0) = 0, the graph must pass through the origin: zero input always gives zero output. That's the test for proportionality — if there's any fixed starting amount, a flat fee, a head start, a base charge, the relationship isn't proportional anymore, even if it still changes at a constant rate.

Linear relationships relax that one condition. They still change at a constant rate, the slope m, but they're allowed to start somewhere else: y = mx + b, with b the value when x = 0. Find m from two points using (y2 - y1)/(x2 - x1), then plug one point back in to solve for b. Every proportional relationship is linear, with b = 0, but most linear relationships are not proportional.

This is the algebra behind reading a graph as a story: a steady hourly wage with no signing bonus is proportional, a phone plan with a monthly fee plus a per-gigabyte rate is linear but not proportional. Once you can build the rule from two points or a table, you're ready to judge how well a line, versus an inverse or exponential curve, actually fits a set of data.