To determine which type of function best models the given data, we can analyze the values of \( x \) and \( y \).
Given the table:
\( x \) | \( y \) ---- | ---- 2 | 0.5 5 | -1.75 8 | -4
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Linear Function: A linear function would show a constant rate of change in \( y \) for equal increments in \( x \).
- Checking the differences:
- From \( x = 2 \) to \( x = 5 \): \( y \) goes from 0.5 to -1.75 (\( \Delta y = -2.25 \))
- From \( x = 5 \) to \( x = 8 \): \( y \) goes from -1.75 to -4 (\( \Delta y = -2.25 \))
- There is a constant change in \( y \) as \( x \) increases, which is a characteristic of a linear function.
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Quadratic Function: A quadratic function would have a second difference that is constant, which typically means the values of \( y \) would change in a parabolic manner. We don’t observe that kind of change here.
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Exponential Function: An exponential function would show a constant multiplicative change, which isn’t evident from the data provided.
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Cubic Function: A cubic function would likely require more points to determine curvature, which might not be observable without more information.
Based on the changes in the values of \( y \) that correspond to consistent increments in \( x \), while we observe negative values and a downward trend, this data best fits a linear function.
Conclusion:
The best type of function to model the data in the table is linear.