Direct vs Inverse Variation

Direct variation and inverse variation are the two simplest non-trivial relationships between variables — and the foundation for understanding more complex models.

Direct variation: y = kx

Two quantities vary directly if $y = k x$ for some non-zero constant $k$ (the constant of variation or constant of proportionality).

• As $x$ doubles, $y$ doubles.
• As $x$ halves, $y$ halves.
• Graph passes through the origin with slope $k$.

Examples: distance vs time at constant speed ($d = v t$), Hooke's law ($F = k x$), simple paychecks ($\text{pay} = \text{rate} \cdot \text{hours}$).

Inverse variation: y = k/x

Two quantities vary inversely if $y = k/x$.

• As $x$ doubles, $y$ halves.
• As $x \to \infty$, $y \to 0$.
• Graph is a hyperbola, never crosses the axes.

Examples: Boyle's law (pressure × volume = constant at constant temp), distance for fixed work ($t = \text{distance} / v$), ohm's law variants.

How to tell which from data

Plot $y$ vs $x$. If the points lie on a straight line through the origin, direct variation. If they lie on a hyperbola decaying to zero, inverse variation. Or check if $\frac{y}{x}$ is constant (direct) vs $xy$ is constant (inverse).

Combined and joint variation

• Joint variation: $y = kxz$ (two direct variables).
• Combined: $y = kx/z$ (one direct, one inverse). Example: gravitational force $F = G m_1 m_2 / r^2$ — direct in masses, inverse-square in distance.

Verdict

Identify by the question "as one increases, does the other increase or decrease, and by what proportion?" Direct → both move together; inverse → opposite direction with reciprocal proportion.