Direct variation describes a simple relationship between two variables. We say $y$ **varies directly** **with **$x$ (or *as* $x$, in some textbooks) if:

$y=kx$

for some constant $k$.

This means that as $x$ increases, $y$ increases and as $x$ decreases, $y$ decreases—and that the ratio between them always stays the same.

The graph of the direct variation equation is a straight line through the origin.

Direct Variation Equation

for $3$ different values of $k$

Inverse variation describes another kind of relationship. We say $y$** varies inversely with **$x$ (or *as* $x$, in some textbooks) if*:*

$xy=k$,

or, equivalently,

$y=\frac{k}{x}$

for some constant $k$.

This means that as $x$ increases, $y$ decreases and as $x$ decreases, $y$ increases.

The graph of the inverse variation equation is a hyperbola.

Inverse Variation Equation

for $3$ different values of $k$