Understanding Proportional Relationships

Two quantities share a proportional relationship if the ratio between them is always exactly the same. Imagine buying apples where each apple costs exactly $2$. Whether you buy $1$ apple or $10$, the ratio of the total cost to the number of apples remains constant.

The Constant of Proportionality

This constant ratio is called the constant of proportionality, usually represented by the letter $k$. You can find $k$ by dividing the $y$-value by the $x$-value:

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

Once you know $k$, you can write the relationship as an algebraic equation:

$y = kx$

Identifying Proportions in Tables

To check if a table shows a proportional relationship, calculate the ratio $\frac{y}{x}$ for every pair of numbers. If the result is the same for every single pair, the relationship is proportional.

Example: Does this table show a proportional relationship?

• $x$: 2, 4, 6
• $y$: 5, 10, 15

Let's check the ratios:

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

Since every ratio equals $2.5$, the table is proportional. The constant of proportionality is $k = 2.5$, and the equation for this table is $y = 2.5x$.

Identifying Proportions on a Graph

When you graph a proportional relationship, the points will always form a straight line that passes directly through the origin $(0, 0)$.

If a line is straight but does not cross through $(0, 0)$, or if the line curves, the relationship is not proportional. To find $k$ from a graph, pick any clear point $(x, y)$ on the line and divide $y$ by $x$.

Identifying Proportions in Equations

A proportional relationship equation always looks like $y = kx$ (without any extra numbers added or subtracted at the end).

Example: In the equation $y = 3x$, what is the constant of proportionality?

Because the equation is in the exact form $y = kx$, the constant $k$ is simply the number multiplying $x$. Therefore, $k = 3$. This means for every $1$ unit $x$ increases, $y$ increases by $3$ units.