To determine whether the data represents a direct variation or an inverse variation, we need to analyze how \( x \) and \( y \) are related.
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Direct Variation: In direct variation, \( y = kx \), where \( k \) is a constant. As \( x \) increases, \( y \) also increases.
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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:
\[ y = \frac{8}{x} \]