Proportional Relationships and the Constant of Proportionality

One quantity is a fixed multiple of the other

Apples cost $2$ dollars each. However many you buy, the cost is always the number of apples times $2$:

$$y = 2x$$

That is a proportional relationship. In general:

$$y = kx$$

The fixed multiplier $k$ is the constant of proportionality. It is the same number for every pair of values, and finding it is what most questions ask for.

Finding k from a table

Divide $y$ by $x$ in each row. Proportional means every row gives the same answer.

Apples ($x$)	Cost ($y$)	$y \div x$
$3$	$6$	$2$
$5$	$10$	$2$
$8$	$16$	$2$

Every row gives $2$, so $k = 2$ and the equation is $y = 2x$.

Here is a table that fails the test:

Hours ($x$)	Cost ($y$)	$y \div x$
$1$	$8$	$8$
$2$	$13$	$6.5$
$3$	$18$	$6$

The quotients change, so this is not proportional. Looking at it, there is a $3$ dollar charge plus $5$ dollars an hour — a fixed fee spoils the constant ratio.

k is the unit rate

$$k = \frac{y}{x}$$

That is the amount of $y$ for one of $x$ — exactly a unit rate. The two names describe the same number seen from different angles: “unit rate” comes from the arithmetic, “constant of proportionality” from the equation.

Situation	$k$	Meaning
$y = 2x$	$2$	$2$ dollars per apple
$y = 55x$	$55$	$55$ miles per hour
$y = 0.5x$	$0.5$	half a cup per serving

The graph is a line through the origin

• y = 2x

Two features identify a proportional relationship on a graph, and both are required.

It is straight. The multiplier $k$ never changes, so every step right of $1$ raises the line by the same $k$. A constant step is what makes a line straight.

It passes through the origin. Zero apples cost zero dollars. Substituting $x = 0$ into $y = kx$ gives $y = 0$ whatever $k$ is, so $(0, 0)$ is on every proportional graph.

Why the origin is the deciding test

A straight line that misses the origin is not proportional, even though it looks similar.

Take a taxi charging $3$ dollars to start plus $2$ per mile:

$$y = 2x + 3$$

That graph is straight, but it crosses the $y$-axis at $3$. Check the ratios:

Miles	Cost	$y \div x$
$1$	$5$	$5$
$2$	$7$	$3.5$
$4$	$11$	$2.75$

The cost per mile keeps falling, because the fixed $3$ dollars is being spread over more miles. There is no single number that turns miles into cost.

A starting amount is exactly what breaks proportionality. This is the difference between $y = kx$ and the general slope-intercept form $y = mx + b$: a proportional relationship is the case where $b = 0$.

What a point on the graph means

Every point $(x, y)$ on a proportional graph reads as ”$x$ of these costs $y$”. Two points are worth naming.

$(1, k)$ is the unit rate made visible — the height of the line above $x = 1$ is $k$ itself. On the apples graph that is $(1, 2)$: one apple, two dollars.

$(0, 0)$ says that none of one means none of the other.

Worked examples

Common mistakes

Practice problems