Question

look at the table below:
which type of relationship exists between x and y?
○ neither
○ direct variation
○ both
○ inverse variation

Explanation:

Step1: Recall direct variation formula

Direct variation is \( y = kx \), where \( k \) is constant. Inverse variation is \( y=\frac{k}{x} \), \( k = xy \) constant.

Step2: Check direct variation (ratio \( \frac{y}{x} \))

For \( x = 1, y = 3 \): \( \frac{y}{x}=\frac{3}{1}=3 \)
For \( x = 2, y = 6 \): \( \frac{y}{x}=\frac{6}{2}=3 \)
For \( x = 3, y = 9 \): \( \frac{y}{x}=\frac{9}{3}=3 \)
For \( x = 4, y = 12 \): \( \frac{y}{x}=\frac{12}{4}=3 \)
Ratio \( \frac{y}{x} \) is constant (\( k = 3 \)), so it's direct variation.

Step3: Check inverse variation (product \( xy \))

\( 1\times3 = 3 \), \( 2\times6 = 12 \), products not constant. So not inverse.

Answer:

Direct variation