To find the constant of proportionality between the two points \((0, 0)\) and \((4, 1)\), we can use the formula for proportionality:
\[ k = \frac{y}{x} \]
where \(k\) is the constant of proportionality, \(y\) is the dependent variable, and \(x\) is the independent variable.
Using the second point \((4, 1)\):
\[ k = \frac{1}{4} \]
This means the constant of proportionality is \(\frac{1}{4}\).
However, since none of the options you provided are \(\frac{1}{4}\), it looks like we may be looking for a negative proportionality or perhaps else.
Let's write the equation of the line formed by these two points. The general equation for a line in slope-intercept form \(y = mx + b\) can help us identify the relationship. The slope \(m\) (which could be thought of as the constant of proportionality in the case of direct proportion) can be found using the two points:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{4 - 0} = \frac{1}{4} \]
This slope indicates that the relationship is proportional with a constant factor of \(\frac{1}{4}\).
Thus, to evaluate the options provided:
- A: 4 - Incorrect
- B: 14 - Incorrect
- C: -4 - Incorrect
- D: -\(\frac{1}{4}\) - Incorrect
None of the options appear to reflect the positive constant of proportionality \(\frac{1}{4}\) derived from the points given.
If we had to choose from these options under normal circumstances, none apply. If you're looking for the negative factor of the line, outputting the slope as negative, the options do not present anything valid from the derived relationship.
If the context allowed for values or interpretations that lead to these choices, please provide more clarification. But on a straightforward assessment of proportionality with the given data, we find it remains as \( \frac{1}{4} \).
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