To determine which function best models the data in the table, we can start by looking for patterns in the values for \( x \) and \( y \).
Let's examine the given data points:
- \( (2, 8.5) \)
- \( (4, 46) \)
- \( (6, 107.5) \)
- \( (8, 193) \)
To identify the type of function, we can calculate the differences in \( y \) values:
- First differences (subtract each \( y \) from the next):
- From \( 8.5 \) to \( 46 \): \( 46 - 8.5 = 37.5 \)
- From \( 46 \) to \( 107.5 \): \( 107.5 - 46 = 61.5 \)
- From \( 107.5 \) to \( 193 \): \( 193 - 107.5 = 85.5 \)
The first differences: \( 37.5, 61.5, 85.5 \)
- Second differences (subtract each first difference from the next):
- From \( 37.5 \) to \( 61.5 \): \( 61.5 - 37.5 = 24 \)
- From \( 61.5 \) to \( 85.5 \): \( 85.5 - 61.5 = 24 \)
The second differences: \( 24, 24 \)
Since the second differences are constant, this suggests that the data follows a quadratic function.
Thus, the best model for the data in the table is quadratic.
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