Lesson 5: Graph Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

1. Using a table: Test for equivalent ratios. For any pair of corresponding quantities $(x, y)$, the ratio $\frac{y}{x}$ must be the same for all non-zero pairs.
2. Using a graph: The graph of the relationship must be a straight line that passes through the origin $(0, 0)$.

Examples

• A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
• A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through $(0,0)$ and $(1,2)$, showing it's proportional.
• A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at $(0,10)$, not the origin, so it is not proportional.

Explanation

Think of it like a recipe. If you double the flour, you must double the sugar. A proportional relationship means two quantities scale up or down together at a steady rate. The graph is a straight line starting from zero because zero input means zero output.

Property

The constant of proportionality, often called the unit rate ($r$), is the constant ratio in a proportional relationship.
If quantities $x$ and $y$ are proportional, their relationship can be described by the equation $y = rx$.
The constant $r$ can be found by calculating the ratio $r = \frac{y}{x}$ for any corresponding pair $(x, y)$ where $x \neq 0$. On a graph, the unit rate is represented by the point $(1, r)$.

Examples

• A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$.
• A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$. The constant of proportionality is 0.55 dollars per pound.
• A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

The constant of proportionality is the 'secret multiplier' that connects the two quantities. It tells you how much of the second quantity you get for exactly one unit of the first quantity, like price per item or miles per hour.