On a line graph, proportional relationships are specifically represented as a line that goes up at a constant rate, and goes through the origin.
Understanding Proportional Relationships Graphically
A proportional relationship exists between two quantities when one quantity is a constant multiple of the other. This constant multiple is known as the constant of proportionality. When plotted on a line graph, this specific type of relationship has two key visual characteristics:
1. It's a Straight Line
The graph must be a line, indicating a steady and predictable change between the two variables plotted on the x and y axes. The relationship isn't curved or broken.
2. It Passes Through the Origin (0,0)
This is a crucial characteristic. The origin is the point where the x-axis and y-axis intersect, representing the point (0,0). A proportional relationship always includes the point (0,0) because if one quantity is zero, the other must also be zero (since one is a constant multiple of the other). If the line doesn't pass through the origin, it represents a linear relationship, but not a proportional one.
3. It Goes Up at a Constant Rate
As the reference states, the line "goes up at a constant rate". This refers to the slope of the line. The slope of the graph of a proportional relationship is constant, meaning the ratio of the change in the y-value to the change in the x-value is always the same. This constant rate is actually the constant of proportionality itself.
- Constant Rate: For every unit increase in the value on the x-axis, the value on the y-axis increases by a fixed amount.
- Unit Rate: The point on the graph where the x-value is 1 (assuming x is the independent variable) represents the unit rate of the relationship. This is the value of the dependent variable (y) when the independent variable (x) is 1.
Visual Summary
| Characteristic | Description | Why it's important for proportionality |
|---|---|---|
| Line Graph | Must be a straight line. | Indicates a constant relationship between variables. |
| Goes Through Origin | Passes through the point (0,0). | Shows that when one quantity is zero, the other is also zero. |
| Constant Rate (Slope) | The steepness of the line is consistent. | Represents the constant of proportionality/unit rate. |
Example of a Proportional Relationship Graph
Imagine a scenario where you earn $15 per hour.
- Variables: Time (hours) on the x-axis, Money Earned ($) on the y-axis.
- Relationship: Money Earned = 15 * Time (hours)
- Constant of Proportionality: 15
When you graph this:
- At 0 hours, you earn $0. This is the point (0,0), which is the origin.
- At 1 hour, you earn $15. This is the point (1,15). (15 is the unit rate).
- At 2 hours, you earn $30. This is the point (2,30).
- At 3 hours, you earn $45. This is the point (3,45).
When you plot these points, they form a straight line that starts at the origin (0,0) and goes up at a constant rate ($15 for every 1 hour).
In summary, on a line graph, a proportional relationship is unmistakable: it is always a straight line that starts precisely at the origin and maintains a constant upward slope.
