A proportional relationship can be identified by the consistency of the ratio \( \frac{y}{x} \) across all pairs of values (excluding \( x = 0 \), where \( y \) should also be \( 0 \)).
Let's analyze each table:
A.
- \( \frac{0}{0} \) (undefined)
- \( \frac{4}{1} = 4 \)
- \( \frac{5}{2} = 2.5 \)
- \( \frac{6}{3} = 2 \)
The ratios are not consistent.
B.
- \( \frac{2}{2} = 1 \)
- \( \frac{4}{4} = 1 \)
- \( \frac{8}{6} = 1.33 \)
- \( \frac{12}{6} = 2 \)
The ratios are not consistent.
C.
- \( \frac{0}{0} \) (undefined)
- \( \frac{3}{1} = 3 \)
- \( \frac{6}{2} = 3 \)
- \( \frac{9}{3} = 3 \)
The ratios are consistent and equal to \( 3 \).
D.
- \( \frac{0}{0} \) (undefined)
- \( \frac{10}{5} = 2 \)
- \( \frac{15}{10} = 1.5 \)
- \( \frac{20}{15} = 1.33 \)
The ratios are not consistent.
Based on this analysis, the only option that represents a proportional relationship is:
C.
\( y = 3x \) (which is a constant multiple of \( x \)).