Question

a) Describe how you know that a graph is proportional.



b) Describe how you know an equation is proportional. Give an example of an equation that is proportional and a graph that is not proportional.



c) Describe how you know that a table is proportional.

Answer 1

### a) How to Determine if a Graph is Proportional

A graph is proportional if it represents a linear relationship between two variables that passes through the origin (0, 0). Specifically, you can determine if a graph is proportional by looking for the following characteristics:

1. **Straight Line**: The graph must be a straight line.
2. **Passes Through the Origin**: The line must intersect the origin (0, 0). This indicates that when one variable is zero, the other variable is also zero.
3. **Constant Rate of Change**: The slope of the line, which represents the ratio of change between the two variables, should remain constant throughout the graph.

### b) How to Determine if an Equation is Proportional

An equation is considered proportional if it can be represented in the form \( y = kx \), where \( k \) is a constant (the rate of change or the constant of proportionality). Key characteristics of a proportional equation include:

1. **Directly Related Variables**: The variables are directly related with no added constants or intercepts.
2. **No Constant Term**: The equation does not include a constant term (i.e., no "+ b" term).
3. **Linear Relationship Through the Origin**: The equation's graph would be a straight line that passes through the origin.

**Example of a Proportional Equation**:
- \( y = 3x \) (This represents a proportional relationship, where \( k = 3 \)).

**Example of a Non-Proportional Equation**:
- \( y = 2x + 1 \) (This is not proportional because it has a constant term, and its graph will not pass through the origin).

### c) How to Determine if a Table is Proportional

A table is proportional if the ratio of the y-values to the corresponding x-values is constant for all pairs of values. To identify a proportional table, you can check the following:

1. **Calculate Ratios**: For each pair of values (x, y), calculate the ratio \( \frac{y}{x} \) (assuming \( x \) is not zero).
2. **Constant Ratios**: All calculated ratios should be the same (a constant).
3. **Origin Inclusion**: If one of the entries in the table corresponds to \( x = 0 \), then the corresponding \( y \) value should also be zero.

**Example of a Proportional Table**:
| x | y |
|----|-----|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |

- Ratios: \( \frac{3}{1} = 3 \), \( \frac{6}{2} = 3 \), \( \frac{9}{3} = 3 \) (constant ratio of 3).

**Example of a Non-Proportional Table**:
| x | y |
|----|-----|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |

- Ratios: \( \frac{2}{1} = 2 \), \( \frac{5}{2} = 2.5 \), \( \frac{8}{3} \approx 2.67 \) (not constant).