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.
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:
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:
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 $\,k\ne 0$