# Direct & Inverse Variation

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
$k$