Inverse Variation Calculator

Explore y = k/x relationships. Find the constant of variation k from a data point and predict y for any x value.

Select Mode & Enter Values

Result

Constant of Variation (k)
20
k 20
Predicted y 2
Equation y = 20/x

Step-by-Step Solution

y = k / x

Understanding Inverse Variation

Inverse variation (also called inverse proportionality) describes a relationship between two variables where their product is constant. When one variable increases, the other decreases proportionally. The mathematical form is y = k / x, where k is the constant of variation.

This relationship is fundamental in physics, economics, and engineering. For example, Boyle's Law states that gas pressure and volume are inversely proportional at constant temperature: P = k / V.

Key Concepts

The Equation y = k/x

k is the constant of variation. When x doubles, y is halved. The product x * y always equals k.

y = k / x, so x * y = k

Finding the Constant k

Given a known pair (x, y), simply multiply them to find k.

k = x * y

Hyperbola Graph

The graph of y = k/x forms a hyperbola with two branches in opposite quadrants.

Asymptotes: x = 0 and y = 0

Comparing Two Points

If (x1, y1) and (x2, y2) are on the same inverse variation curve, then x1 * y1 = x2 * y2.

x1 * y1 = x2 * y2 = k

Inverse Variation vs. Direct Variation

In direct variation (y = kx), both variables increase or decrease together. In inverse variation (y = k/x), one increases while the other decreases. Direct variation produces a straight line through the origin; inverse variation produces a hyperbola.

Real-World Examples

  • Speed and Time: For a fixed distance, increasing speed decreases travel time (d = speed * time).
  • Workers and Days: More workers means fewer days to complete a fixed amount of work.
  • Boyle's Law: At constant temperature, gas pressure and volume are inversely proportional.
  • Gear Ratios: Larger gears turn slower, smaller gears turn faster for the same input.
  • Electrical Resistance: For a fixed voltage, current and resistance are inversely related (Ohm's Law: I = V/R).

Properties of the Hyperbola y = k/x

The graph of an inverse variation equation has several distinctive features. The curve never crosses either axis (the axes are asymptotes). For positive k, the curve exists in the first and third quadrants. For negative k, it exists in the second and fourth quadrants. The curve is symmetric about the line y = x and about the line y = -x.

How to Use This Calculator

  1. Find k from Data Point: Enter a known (x, y) pair and a new x to predict y.
  2. Predict y: Enter a known k and x to find y = k/x.
  3. Predict x: Enter a known k and y to find x = k/y.