To determine which type of function best models the relationship between \( x \) and \( y \) from the given data points:
\[ \begin{align*} (2, 2) \ (5, -2) \ (8, -6) \ \end{align*} \]
Let's analyze the changes in \( y \) as \( x \) changes:
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From \( x = 2 \) to \( x = 5 \):
- Change in \( y \): \( -2 - 2 = -4 \)
- Change in \( x \): \( 5 - 2 = 3 \)
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From \( x = 5 \) to \( x = 8 \):
- Change in \( y \): \( -6 - (-2) = -4 \)
- Change in \( x \): \( 8 - 5 = 3 \)
The changes are consistent (the first change is -4 over a change in \( x \) of 3; the second change is also -4 over the same change in \( x \)). This indicates a potential linear relationship, as there is a constant rate of change.
To further confirm, we can calculate the slope (\( m \)): \[ m = \frac{\Delta y}{\Delta x} = \frac{-4}{3} \]
Since the changes in \( y \) for equal changes in \( x \) are constant, the relationship between \( x \) and \( y \) is linear.
Thus, the function that best models the data in the table is linear.