Question

Show a proportional graph and equation and compare to a non proportional graph and equation.
Proportional: y =

Answer 1

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**:
1. When \(x = 0\), \(y = 2(0) = 0\)
2. When \(x = 1\), \(y = 2(1) = 2\)
3. When \(x = 2\), \(y = 2(2) = 4\)
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**:
1. When \(x = 0\), \(y = 2(0) + 3 = 3\)
2. When \(x = 1\), \(y = 2(1) + 3 = 5\)
3. When \(x = 2\), \(y = 2(2) + 3 = 7\)
4. 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
1. **Proportional**:
- **Equation**: \(y = 2x\)
- **Graph**: Passes through (0,0)
- **Constant Ratio**: The ratio of \(y\) to \(x\) is constant (2 in this case).

2. **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.