Direct variation
We say $y$ varies directly as $x$, written $y \propto x$, when $y$ increases at the same rate as $x$. Introducing the constant of proportionality $k$ turns the proportion into an equation:
Key formula
$y \propto x \ \Rightarrow\ y = kx, \qquad k = \frac{y}{x}$
The graph of $y$ against $x$ is a straight line through the origin with gradient $k$.
Finding the constant $k$
Substitute one known pair of values to find $k$, then use $y = kx$ to answer any further question.
Example
Given $y \propto x$ and $y = 12$ when $x = 3$: $k = \frac{12}{3} = 4$, so $y = 4x$. When $x = 7$, $y = 4(7) = 28$.
Powers: $y \propto x^n$
Direct variation also covers powers, such as $y \propto x^2$ or $y \propto \sqrt{x}$. The method is the same — write $y = kx^n$, find $k$ from a known pair, then substitute.
Example
Given $y \propto x^2$ and $y = 18$ when $x = 3$: $k = \frac{18}{3^2} = \frac{18}{9} = 2$, so $y = 2x^2$. When $x = 5$, $y = 2(25) = 50$.
Remember
In direct variation, if $x$ is doubled then $y$ is also doubled (for $y \propto x$). The ratio $\frac{y}{x}$ stays constant.