Lesson 4: Representing Proportional Relationships with Graphs

Property

When graphed, the relationship between two proportional variables has two key characteristics:

1. The graph is a straight line.
2. The graph passes through the origin, which is the point $(0, 0)$.

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

• A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
• The graph of a monthly bus pass cost is a horizontal line at $y=50$. This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
• A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$. The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.

Explanation

Think of a proportional graph as a perfectly straight ramp that starts right at the ground. It is straight because the steepness (the rate) never changes, and it starts at $(0,0)$ because zero input means zero output, like working 0 hours earns 0 dollars.

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.