Inverse variation
We say $y$ varies inversely as $x$, written $y \propto \frac{1}{x}$, when $y$ decreases as $x$ increases (and vice versa) so that their product stays constant. With the constant of proportionality $k$:
Key formula
$y \propto \frac{1}{x} \ \Rightarrow\ y = \frac{k}{x}, \qquad k = xy$
Because $k = xy$, the product of $x$ and $y$ is always the same value $k$.
Finding $k$ and using it
Example
Given $y \propto \frac{1}{x}$ and $y = 4$ when $x = 3$: $k = xy = 3 \times 4 = 12$, so $y = \frac{12}{x}$. When $x = 6$, $y = \frac{12}{6} = 2$.
Powers: $y \propto \frac{1}{x^n}$
The idea extends to powers, such as $y \propto \frac{1}{x^2}$. Then $y = \frac{k}{x^2}$ and $k = x^2 y$.
Example
Given $y \propto \frac{1}{x^2}$ and $y = 4$ when $x = 2$: $k = 2^2 \times 4 = 16$, so $y = \frac{16}{x^2}$. When $x = 4$, $y = \frac{16}{16} = 1$.
Remember
For $y \propto \frac{1}{x}$, doubling $x$ halves $y$. The graph of $y$ against $\frac{1}{x}$ is a straight line through the origin.