﻿ Direct and Inverse Variation

# 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 $\,\frac 1k\ne 0$ (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 $\,k\ne 0$