Constant of proportionality

Where k sits

$$y = kx \qquad\Longleftrightarrow\qquad k = \frac{y}{x}$$

For apples at $2$ dollars each, $y = 2x$ and $k = 2$.

Finding it from a table

Divide $y$ by $x$ in every row. Proportional means they all agree.

$x$	$y$	$y \div x$
$3$	$6$	$2$
$5$	$10$	$2$
$8$	$16$	$2$

One row is never enough — any single pair produces some quotient. What matters is that they match, which is the test set out in proportional relationships.

What k looks like on a graph

The graph of $y = kx$ is a straight line through the origin, and $k$ is how steeply it rises. The point $(1, k)$ puts the constant on the picture directly: the height of the line above $x = 1$ is $k$.

The line passes through $(0, 0)$ because substituting $x = 0$ gives $y = 0$ whatever $k$ is. That is the test that separates $y = kx$ from the general slope-intercept form $y = mx + b$ — a proportional relationship is the case where $b = 0$, with no starting amount.

The same idea in three vocabularies

Setting	Name for k
arithmetic	unit rate
a proportional equation	constant of proportionality
a graph or function	slope, or rate of change

A question that asks for any one of these is asking for the same number.