Direct Variation

Definition

Direct variation describes a functional relationship where two variables increase or decrease together at a constant ratio.

• How to read: “y equals k times x.”
• Meaning: As $x$ grows, $y$ scales proportionally; the ratio $y/x = k$ is constant.

Why It Matters

Assuming direct variation in systems where it applies allows for predictable scaling. It’s the simplest mathematical relationship, governing everything from wage calculations to linear physical laws like Hooke’s Law.

Core Concepts

• Equation: $y = kx$, where $k$ is the constant of proportionality.
• The Constant $k$: $k \neq 0$ is a constant determined using initial conditions (a known data point).
  • How to read: “Constant of proportionality k.”
  • Meaning: The fixed ratio linking variables.
• Graph: The graph of a direct variation is a straight line passing through the origin $(0,0)$.
• Joint Variation ($z = kxy$): $z$ is directly proportional to the product of two or more variables.

Connected Concepts

• Inverse Variation
• Linear Functions
• Scale