Direct and Inverse Variation

Direct Variation

$y$ varies directly with $x$ if: $y = kx$

where $k$ is the constant of variation.

Characteristics:

• When $x$ doubles, $y$ doubles
• When $x$ triples, $y$ triples
• Graph passes through the origin
• Graph is a straight line

Example: If $y = 6$ when $x = 2$: $6 = k(2)$ → $k = 3$ So $y = 3x$

Inverse Variation

$y$ varies inversely with $x$ if: $y = \frac{k}{x}$ or $xy = k$

Characteristics:

• When $x$ doubles, $y$ is halved
• When $x$ increases, $y$ decreases
• Graph is a hyperbola

Example: If $y = 8$ when $x = 3$: $8 = \frac{k}{3}$ → $k = 24$ So $y = \frac{24}{x}$

Real-World Examples

Direct: Distance and time (at constant speed) Inverse: Speed and time (for fixed distance)