To determine which type of function best models the data in the table, we should first analyze the values of \(x\) and \(y\). Based on the representation, it looks like we might be missing some details in the table. Let's consider that for each \(x\), the corresponding \(y\) values given are:
- For \(x = 2\), \(y = 8.5\)
- For \(x = 4\), \(y = 46\)
- For \(x = 6\), \(y = 107.5\)
- For \(x = 8\), \(y = 193\)
Now, let's check how the values of \(y\) change as \(x\) increases:
- The increase from \(y = 8.5\) to \(y = 46\) is \(46 - 8.5 = 37.5\).
- The increase from \(y = 46\) to \(y = 107.5\) is \(107.5 - 46 = 61.5\).
- The increase from \(y = 107.5\) to \(y = 193\) is \(193 - 107.5 = 85.5\).
The increments in \(y\) (37.5, 61.5, 85.5) are not constant, which indicates that the relationship is not linear.
Next, let’s check if the second differences (the increases in the increments) are consistent:
- From \(37.5\) to \(61.5\) is an increase of \(61.5 - 37.5 = 24\).
- From \(61.5\) to \(85.5\) is an increase of \(85.5 - 61.5 = 24\).
Since the second differences are constant, this suggests that the relationship is quadratic.
Thus, based on the changes in \(y\) with respect to \(x\) and observing the second differences, the best model for the data is quadratic.