# Getting Bigger? Getting Smaller? (Direct and Inverse Variation)

*Bigger* means
*farther away from zero*
and
*smaller* means
*closer to zero*.
(This is discussed in more detail in a future section.)

Suppose that $\,y = 2x\,.$ When $\,x\,$ gets bigger, $\,y\,$ gets bigger. When $\,y\,$ gets bigger, $\,x\,$ gets bigger. In this type of relationship, $\,x\,$ and $\,y\,$ ‘follow each other’ in size: when one gets bigger, so does the other. When one gets smaller, so does the other.

This kind of relationship between two variables is called
*direct variation*:
if there is a nonzero number $\,k\,$ for which $\,y = kx\,,$
then we say that
‘$\,y\,$ varies directly as $\,x\,$’.

Now suppose that $\,y = \frac{2}{x}\,.$ When $\,x\,$ gets bigger, $\,y\,$ gets smaller. When $\,x\,$ gets smaller, $\,y\,$ gets bigger. In this type of relationship, $\,x\,$ and $\,y\,$ have sizes that go in different directions: when one gets bigger, the other gets smaller. When one gets smaller, the other gets bigger.

This kind of relationship between two variables
is called *inverse variation*:
if there is a nonzero number $\,k\,$
for which $\displaystyle \,y = \frac{k}{x}\,,$
then we say that
‘$\,y\,$ varies inversely as $\,x\,$’.

## Examples

Intuition: Both variables are ‘upstairs’ on opposite sides of the equation.

Intuition: One variable is ‘upstairs’ and the other ‘downstairs’ on opposite sides of the equation.