Direct and Inverse Variation

The earlier lesson Getting Bigger? Getting Smaller? introduces the concepts of direct and inverse variation.
Study it first, being sure to click-click-click several exercises at the bottom to check your understanding.
This current lesson builds on these prior concepts.

DIRECT VARIATION equivalent statements

The following are equivalent:

$y = kx\,$, for $\,k\ne 0$
$y\,$ varies directly as $\,x$
$y\,$ is directly proportional to $\,x$
$y\,$ is proportional to $\,x$

The nonzero constant $\,k\,$ is called the constant of proportionality.

When you are told that ‘$y\,$ is proportional to $x\,$’ then direct variation is being described.
To emphasize this fact, the extra word ‘directly’ can be inserted: ‘$y\,$ is directly proportional to $\,x$’.

Recall that variables that are directly proportional ‘follow each other in size’:
when one gets bigger (farther away from zero), so does the other;
when one gets smaller (closer to zero), so does the other.

Notice: if $\,y\,$ is proportional to $\,x\,$, then $\,x\,$ is proportional to $\,y$.
Thus, we can simply say ‘$x\,$ and $\,y$ are proportional’ or ‘$y\,$ and $\,x$ are proportional’.
Why is this? Study the following list of equivalent sentences:

$y\,$ is proportional to $\,x\,$	given; assumed to be true
$y = kx\,$, for $\,k\ne 0$	an equivalent statement of direct variation; see the list above
$x = \frac 1k y\,$, for $\,\frac 1k\ne 0$	multiplication property of equality (divide both sides by $\,k\ne 0\,$); also, $\,k\,$ is nonzero if and only if its reciprocal is nonzero
$x\,$ is proportional to $\,y\,$	an equivalent statement of direct variation; see the list above

Also notice:

if two variables are proportional, then whenever one of the variables is zero, the other must also be zero
the graph of the relationship between two directly proportional variables (i.e., the graph of $\,y = kx\,$ for $\,k\ne 0\,$)
is a non-horizontal, non-vertical line that passes through the origin

Next, we talk about inverse variation:

INVERSE VARIATION equivalent statements

The following are equivalent:

$\displaystyle y = \frac{k}{x}\,$, for $\,k\ne 0$
$y\,$ varies inversely as $\,x$
$y\,$ is inversely proportional to $\,x$

Recall that variables that are inversely proportional have sizes that ‘go in opposite directions’:
when one gets bigger, the other gets smaller;
when one gets smaller, the other gets bigger.

Similar to the argument above: if $\,y\,$ is inversely proportional to $\,x\,$, then $\,x\,$ is inversely proportional to $\,y$.
Thus, we can simply say ‘$x\,$ and $\,y$ are inversely proportional’ or ‘$y\,$ and $\,x$ are inversely proportional’.

Also notice:

if two variables are inversely proportional, then neither variable can equal zero
the graph of the relationship between two inversely proportional variables (i.e., the graph of $\,y = \frac{k}{x}\,$ for $\,k\ne 0\,$)
is a vertical scaling of the reciprocal function

Sometimes it is necessary to talk about a relationship between more than two variables:

JOINT PROPORTIONALITY equivalent statements

The following are equivalent:

$\displaystyle z = kxy\,$, for $\,k\ne 0$
$z\,$ varies jointly as $\,x\,$ and $\,y$
$z\,$ is jointly proportional to $\,x\,$ and $\,y\,$

By combining phrases, a wide variety of relationships can be described between variables.
For example:

‘$z\,$ is proportional to $\,x\,$ and inversely proportional to $\,y$’
is equivalent to
$\displaystyle z = \frac{kx}{y}\,$ for $\,k\ne 0\,$
‘$w\,$ varies inversely as the square of $\,t\,$, and directly as the square root of $\,\ell$’
is equivalent to
$\displaystyle w = \frac{k\sqrt{\ell}}{t^2}\,$ for $\,k\ne 0$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Proportionality Problems

DIRECT AND INVERSE VARIATION