To determine which of the tables represents a proportional relationship, we need to check if the ratio \( \frac{y}{x} \) is constant for all values of \( x \) (except when \( x = 0 \), as this will cause the ratio to be undefined).
Let's analyze each option:
A.
- For \( x = 0, y = 0 \): ratio is undefined.
- For \( x = 1, y = 4 \): \( \frac{4}{1} = 4 \).
- For \( x = 2, y = 5 \): \( \frac{5}{2} = 2.5 \).
- For \( x = 3, y = 6 \): \( \frac{6}{3} = 2 \).
The ratios are not constant, so this is not a proportional relationship.
B.
- For \( x = 0, y = 0 \): ratio is undefined.
- For \( x = 2, y = 4 \): \( \frac{4}{2} = 2 \).
- For \( x = 4, y = 8 \): \( \frac{8}{4} = 2 \).
- For \( x = 6, y = 12 \): \( \frac{12}{6} = 2 \).
The ratio is constant, equal to 2, so this is a proportional relationship.
C.
- For \( x = 0, y = 0 \): ratio is undefined.
- For \( x = 1, y = 3 \): \( \frac{3}{1} = 3 \).
- For \( x = 2, y = 6 \): \( \frac{6}{2} = 3 \).
- For \( x = 3, y = 9 \): \( \frac{9}{3} = 3 \).
The ratio is constant, equal to 3, so this is a proportional relationship.
D.
- For \( x = 0, y = 0 \): ratio is undefined.
- For \( x = 5, y = 10 \): \( \frac{10}{5} = 2 \).
- For \( x = 10, y = 15 \): \( \frac{15}{10} = 1.5 \).
- For \( x = 15, y = 20 \): \( \frac{20}{15} = \frac{4}{3} \).
The ratios are not constant, so this is not a proportional relationship.
Thus, the tables that represent a proportional relationship are B and C.