Sure! Let’s explore a proportional relationship and a non-proportional relationship along with their corresponding equations.
Proportional Relationship
In a proportional relationship, the ratio between the two variables (usually \(x\) and \(y\)) is constant. The graph of a proportional relationship is a straight line that passes through the origin (0,0).
Equation: \[ y = kx \] where \(k\) is the constant of proportionality.
For example, let’s take \(k = 2\): \[ y = 2x \]
Graph:
- When \(x = 0\), \(y = 2(0) = 0\)
- When \(x = 1\), \(y = 2(1) = 2\)
- When \(x = 2\), \(y = 2(2) = 4\)
- When \(x = 3\), \(y = 2(3) = 6\)
The points (0,0), (1,2), (2,4), and (3,6) will create a straight line when plotted on a graph.
Non-Proportional Relationship
In a non-proportional relationship, the ratio between \(x\) and \(y\) is not constant. The graph of a non-proportional relationship does not pass through the origin.
Equation: \[ y = kx + b \] where \(b\) is a constant.
For example, let's take \(k = 2\) and \(b = 3\): \[ y = 2x + 3 \]
Graph:
- When \(x = 0\), \(y = 2(0) + 3 = 3\)
- When \(x = 1\), \(y = 2(1) + 3 = 5\)
- When \(x = 2\), \(y = 2(2) + 3 = 7\)
- When \(x = 3\), \(y = 2(3) + 3 = 9\)
The points (0,3), (1,5), (2,7), and (3,9) will create a straight line that does not pass through the origin.
Comparison
-
Proportional:
- Equation: \(y = 2x\)
- Graph: Passes through (0,0)
- Constant Ratio: The ratio of \(y\) to \(x\) is constant (2 in this case).
-
Non-Proportional:
- Equation: \(y = 2x + 3\)
- Graph: Does not pass through (0,0) (it starts at (0,3))
- Changing Ratio: The ratio of \(y\) to \(x\) changes as \(x\) increases.
In summary, while proportional relationships have a constant rate of change and pass through the origin, non-proportional relationships have a different starting point (given by the constant \(b\)) and do not maintain a constant ratio.