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.