A proportional relationship is one in which the ratio between the two variables is constant, and the graph of such a relationship passes through the origin (0, 0).

Let's analyze each option:

1. A graph of a line that passes through the points (0, -1) and (1, -5):
   This line does not pass through the origin (0, 0). For it to represent a proportional relationship, it must pass through the origin, so this is not a proportional relationship.

2. A graph of a line that passes through the points (0, 0) and (4, -1):
   This line passes through the origin, but the ratio between the points is not constant. The slope is $\frac{-1-0}{4-0} = \frac{-1}{4}$, which is not a simple ratio that would indicate proportionality (it represents a negative linear relationship, but not a constant proportional relationship). This line is not proportional.

3. A graph of a line that passes through the points (0, 4) and (1, 4):
   This line is horizontal, meaning the y-values are constant at 4. Since proportional relationships involve the ratio between $y$ and $x$ being constant, this graph does not represent a proportional relationship, as the graph is flat and does not pass through the origin.

4. A graph of a line that passes through the points (0, 1) and (-1, -3):
   This line does not pass through the origin. It also does not have a constant ratio of $y$ to $x$ because it has a slope of $\frac{-3-1}{-1-0} = \frac{-4}{-1} = 4$, which is a linear relationship but not proportional.

Correct Answer:

None of these graphs represent a proportional relationship because for a proportional relationship to exist, the graph must pass through the origin, and the ratio of the coordinates must be constant.

Let me know if you'd like further clarification on any of the concepts!

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Here are some related questions to help deepen your understanding:

1. What is the definition of a proportional relationship in mathematical terms?
2. How do you determine the slope of a line given two points?
3. What is the difference between a proportional relationship and a non-proportional linear relationship?
4. How can you use the equation of a line to determine if it represents a proportional relationship?
5. What are the characteristics of a graph that represents a proportional relationship?

Tip: Always remember that proportional relationships pass through the origin, and the slope (rate of change) remains constant between any two points on the line.