To determine which type of function best models the data, we need to observe how the values of \(y\) change as \(x\) increases. Let's analyze the differences between the \(y\) values:
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Calculate the differences in \(y\):
- 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 are: \(37.5, 61.5, 85.5\).
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Calculate the second differences:
- 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 are constant: \(24\).
Since the second differences are constant, this indicates that the relationship between \(x\) and \(y\) is best modeled by a quadratic function.
Thus, the best answer from the options provided is:
quadratic.