Graphing Proportional Variables

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.