To determine which type of function best fits the data provided in the table, let's analyze the values of \( y \) as \( x \) increases.
The data points given are:
\[ \begin{align*} x & : 2, 4, 6, 8 \ y & : 8.5, 46, 107.5, 193 \ \end{align*} \]
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Calculate the differences of \( y \):
\[ \begin{align*} y(4) - y(2) & : 46 - 8.5 = 37.5 \ y(6) - y(4) & : 107.5 - 46 = 61.5 \ y(8) - y(6) & : 193 - 107.5 = 85.5 \ \end{align*} \]
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Calculate the differences of the first differences:
\[ \begin{align*} 61.5 - 37.5 & = 24 \ 85.5 - 61.5 & = 24 \ \end{align*} \]
The second differences are constant (24), indicating that this data can best be modeled by a quadratic function, since the second differences are constant.
Therefore, the best model for the data is:
Quadratic.