Building Functions for Inversely Proportional Relations (F 602)

Some pairs of quantities trade off so that their product never changes. Painters and hours work this way on a fixed job: more hands means less time, because the total amount of work (painter-hours) stays constant. If you double the crew, each painter still has to do their share, but now there are twice as many people splitting it, so the time gets cut in half. That constant product is the whole idea behind inverse variation, and it turns into the rule y = k over x once you let k stand for that fixed product.

To build the rule, use one known pair of values to solve for k, then plug k back in to answer any other question about the relationship. If y varies inversely with x and y = 12 when x = 5, the product xy is always 60, so k = 60 and y = 60/x. At x = 8, y = 60/8 = 7.5. The same logic handles the painters: 4 painters taking 15 hours means the job takes 4 x 15 = 60 painter-hours total, so 6 painters need 60/6 = 10 hours.

This sits next to linear functions in the toolkit of rules for matching a graph to a story, and the two are easy to confuse because both describe one quantity changing in response to another. The test is whether the two quantities add to a constant or multiply to one. Inverse variation can also stretch: y = k/x² falls even faster as x grows, since doubling x cuts y to a quarter instead of a half, giving a steeper hyperbola near the axes.