Question

18
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) Describe how you know that a graph is proportional.

A graph is considered proportional if it represents a linear relationship that passes through the origin (0,0). This means that as one variable increases, the other variable changes at a constant rate, resulting in a straight line. Key characteristics of a proportional graph include:

- **Straight Line**: The graph must be a straight line.
- **Passes Through the Origin**: The line should intersect the origin, indicating that when one variable is zero, the other variable is also zero.
- **Constant Ratio**: The ratio between the two variables remains constant (the slope of the line).

### 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.

An equation is considered proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant. This demonstrates a direct relationship between \( x \) and \( y \) such that \( y \) is always a constant multiple of \( x \). The key features of a proportional equation are:

- **Direct Variation**: It illustrates that \( y \) varies directly with \( x \).
- **No Constant Term**: The equation should not include a constant term (like \( y = kx + b \) where \( b \neq 0 \)).

**Example of a proportional equation**:
\[ y = 3x \] (where \( k = 3 \))

**Example of a graph that is not proportional**:
The equation \( y = 2x + 5 \) is not proportional because it has a constant term (+5), which means the line does not pass through the origin.

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

A table is proportional if the ratio of the values of one variable to the other variable remains constant throughout the table. To determine if a table is proportional, you can:

- **Calculate Ratios**: For each pair of values, calculate the ratio \( \frac{y}{x} \). If this ratio is the same for all pairs, the table is proportional.
- **Zero in the Origin**: If both variables are zero (i.e., \( x = 0 \) leads to \( y = 0 \)), it strengthens the argument that the relationship is proportional.

**Example of a proportional table**:

| x | y | Ratio \( \frac{y}{x} \) |
|---|----|--------------------------|
| 0 | 0 | N/A |
| 1 | 2 | 2 |
| 2 | 4 | 2 |
| 3 | 6 | 2 |

In this example, the ratio remains constant at 2 throughout, indicating a proportional relationship.