Question

14.
Does the data in the table represent a direct variation or an inverse variation? Write an equation to model the data in the table.

x 2 4 8 12
y 4 2 1 2/3

Answer 1

To determine whether the data represents a direct variation or an inverse variation, we need to analyze how \( x \) and \( y \) are related.

1. **Direct Variation:** In direct variation, \( y = kx \), where \( k \) is a constant. As \( x \) increases, \( y \) also increases.

2. **Inverse Variation:** In inverse variation, \( y = \frac{k}{x} \), where \( k \) is a constant. As \( x \) increases, \( y \) decreases.

Let's analyze the given data:

- When \( x = 2 \), \( y = 4 \)
- When \( x = 4 \), \( y = 2 \)
- When \( x = 8 \), \( y = 1 \)
- When \( x = 12 \), \( y = \frac{2}{3} \)

To check for inverse variation, we can calculate \( k \) by multiplying \( x \) and \( y \):

- For \( x = 2 \), \( y = 4 \) → \( k = 2 \cdot 4 = 8 \)
- For \( x = 4 \), \( y = 2 \) → \( k = 4 \cdot 2 = 8 \)
- For \( x = 8 \), \( y = 1 \) → \( k = 8 \cdot 1 = 8 \)
- For \( x = 12 \), \( y = \frac{2}{3} \) → \( k = 12 \cdot \frac{2}{3} = 8 \)

Since the value of \( k \) is constant (\( k = 8 \)) for all pairs of \( (x, y) \), we can conclude that this data represents an inverse variation.

The equation that models the data is:

\[
y = \frac{8}{x}
\]