AP® Precalculus · Unit 1 · Topic 1.14
Rational Models and Inverse Proportion
Inverse proportion means one quantity equals a constant divided by another, so as the input grows the output shrinks toward zero, and these relationships graph as rational functions with asymptotes that reveal long-run behavior.
When two quantities vary inversely, their relationship has the form y = k/x, or for inverse-square relationships, y = k/x2. The constant k sets the strength of the relationship; the exponent on x controls how fast the effect fades. Light intensity, gravitational force, and electric force all drop off as inverse-square: double the distance and the effect doesn't halve, it quarters, because you divide by distance squared. That's why a lamp looks dramatically dimmer just a few steps away — you're dividing by a fast-growing number.
Graph y = k/x2 and you get a curve that plunges toward zero as x grows, with a vertical asymptote at x = 0 where the function blows up. This shape is just a rational function: a ratio of a constant to a power of x. The bigger the exponent in the denominator, the sharper the initial drop and the flatter the long-run tail. Recognizing k/x or k/x2 patterns lets you predict behavior at extremes without recomputing every point.
Average-cost models use a related but different rational structure: C(x) = (F + vx)/x, a fixed cost F plus variable cost vx, all divided by units x. Split this into F/x + v. As x grows without bound, F/x shrinks toward zero — the fixed cost gets spread so thin it barely matters per unit — leaving C(x) approaching v, the per-unit variable cost. That horizontal asymptote y = v acts as a floor: average cost can get close to v but never drop below it, no matter how much you produce. This connects inverse proportion directly to end behavior and asymptotes, the throughline of the unit.
Key terms
- Inverse variation
- A relationship y = k/x where y decreases as x increases, with k a nonzero constant.
- Inverse-square variation
- A relationship y = k/x2 where the output shrinks with the square of the input.
- Horizontal asymptote
- A horizontal line that a function's output approaches but never reaches as x grows without bound.
- Average cost
- Total cost divided by number of units produced, often modeled as C(x) = (F + vx)/x.
- Fixed cost
- A cost that stays constant no matter how many units are produced.
Watch it explained
- How it worksHow does an inverse square relationship turn into a rational function on a graph?
- Worked exampleFind the gravitational force between two 1 kg masses at 1 m, 2 m, and 4 m apart.
- Going furtherWhat happens to average cost per unit as production climbs without bound?
- Ties togetherHow does the horizontal asymptote of an average-cost model act as a floor on price?
Check yourself
End of topic.