# DIRECT AND INVERSE VARIATION

by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)

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

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
 1 2
AVAILABLE MASTERED IN PROGRESS