Lesson 3: The Constant of Proportionality

Property

In a proportional relationship, the ratio between two quantities, $y$ and $x$, is always constant. This constant value is called the constant of proportionality, represented by the letter $k$. It is also known as the unit rate.

Property

For a proportional relationship, every point $(x, y)$ on the graph represents a specific instance of that relationship. The value $x$ is the amount of the first quantity, and $y$ is the corresponding amount of the second quantity.
Two points have special meaning:

• The point $(0, 0)$ represents the origin, meaning that zero units of the first quantity corresponds to zero units of the second.
• The point $(1, r)$ represents the unit rate, where $r$ is the constant of proportionality.

Examples

• On a graph showing cost vs. gallons of gas, the point $(10, 35)$ means that 10 gallons of gas cost 35 dollars.
• On the same gas graph, the point $(0, 0)$ signifies that if you buy 0 gallons of gas, your cost is 0 dollars.
• On that same graph, the point $(1, 3.5)$ reveals the unit rate: the cost is 3.50 dollars per gallon.

Explanation

Each point on the graph tells a part of the story. The x-coordinate is the 'if' and the y-coordinate is the 'then'. The point $(0,0)$ is the starting point, and the point $(1,r)$ is the key, revealing the rate for just one item.