Introduction to Proportional Relationships

A proportional relationship occurs when two quantities change together at a constant rate. No matter how much the quantities increase or decrease, the ratio between them always stays exactly the same.

The Constant of Proportionality

If you have two quantities, $x$ and $y$, they are proportional if the ratio of $y$ to $x$ is always a constant value. We call this value the constant of proportionality, often represented by the letter $k$.

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

You can also write this relationship as an equation:

$y = kx$

For example, in the equation $y = 3x$, the constant of proportionality is $3$. This means that for every $1$ unit of $x$, $y$ increases by $3$.

Identifying Proportional Relationships in Tables

To check if a table shows a proportional relationship, divide the $y$-value by the $x$-value for every row. If you get the same number every time, the relationship is proportional!

Example: Does this table show a proportional relationship?

$x$	$y$
2	5
4	10
6	15

Let's check the ratios ($\frac{y}{x}$):

• $\frac{5}{2} = 2.5$
• $\frac{10}{4} = 2.5$
• $\frac{15}{6} = 2.5$

Since the ratio is always $2.5$, yes, this is a proportional relationship! The constant of proportionality is $k = 2.5$.

Identifying Proportional Relationships on Graphs

You can easily spot a proportional relationship on a coordinate plane by looking for two key features. The graph must be:

1. A straight line.
2. Passing exactly through the origin $(0, 0)$.

If a line is straight but does not go through $(0,0)$, or if it curves, the relationship is not proportional.

When you graph a proportional relationship like $\frac{y}{x} = 4$ (which is the same as $y = 4x$), the line will start at the origin $(0,0)$ and go up $4$ units on the $y$-axis for every $1$ unit it moves right on the $x$-axis.