Recognizing a proportional relationship in a graph is straightforward if you know what specific visual cues to look for. A graph represents a proportional relationship only if it is a straight line and passes through the origin (0,0).
The Two Essential Checks
Based on the provided reference, identifying a proportional relationship on a graph requires verifying two crucial conditions. Both conditions must be met for the relationship to be considered proportional.
Check 1: Is the Graph a Straight Line?
The first step in recognizing a proportional relationship is to examine the shape of the graph. A proportional relationship between two variables will always result in a straight line when plotted on a coordinate plane.
As stated in the reference: "Check whether the graph is a straight line. If the graph is not a straight line, it does not represent a proportional relationship."
- Why it matters: Proportional relationships have a constant rate of change (represented by the constant of proportionality). This constant rate means the graph's slope is uniform, resulting in a straight line.
Check 2: Does the Line Pass Through the Origin (0,0)?
The second equally important step is to see if the straight line goes through the point where the x-axis and y-axis intersect – the origin (0,0).
According to the reference: "Check whether the line goes through the origin. If the graph does not go through the origin, it does not represent a proportional relationship."
- Why it matters: A proportional relationship means that when one quantity is zero, the other quantity must also be zero. For example, if you buy 0 apples, the cost is $0. On a graph, this corresponds to the point (0,0).
Visual Summary: Proportional vs. Non-Proportional Graphs
Here's a quick way to summarize the checks:
| Graph Characteristic | Proportional Relationship? | Notes |
|---|---|---|
| Straight Line | Required | If not straight, it's not proportional. |
| Passes Through Origin | Required | If straight but misses origin, not proportional. |
What Happens If a Check Fails?
If a graph fails either of these two checks, it does not represent a proportional relationship.
- If the graph is not a straight line: It represents a relationship where the rate of change is not constant (e.g., quadratic, exponential). This is not proportional.
- If the graph is a straight line but does NOT go through the origin: It represents a linear relationship, but specifically not a proportional one. An example is
y = x + 5. This is a straight line, but when x is 0, y is 5 (it passes through (0,5), not (0,0)).
To recognize a proportional relationship in a graph, you must confirm both that it is a straight line and that it passes through the origin.
