To determine which type of function best models the data in the table, we can examine how the values of \( y \) change as \( x \) increases.
Here are the \( x \) and \( y \) values from the table:
\[ \begin{align*} x & : 2, 4, 6, 8 \ y & : 8.5, 46, 107.5, 193 \ \end{align*} \]
Step 1: Check for Linear Growth
If it were linear, the change in \( y \) values would be constant per unit increase in \( x \). Let's calculate the differences in \( y \):
- \( y(4) - y(2) = 46 - 8.5 = 37.5 \)
- \( y(6) - y(4) = 107.5 - 46 = 61.5 \)
- \( y(8) - y(6) = 193 - 107.5 = 85.5 \)
The differences are not constant, indicating that the relationship is not linear.
Step 2: Check for Quadratic Growth
For a quadratic relationship, the second differences should be constant. Let's find the first differences:
- First differences: \( 37.5, 61.5, 85.5 \)
Now calculate the second differences:
- \( 61.5 - 37.5 = 24 \)
- \( 85.5 - 61.5 = 24 \)
The second differences are constant (both are 24), suggesting that the data follows a quadratic model.
Step 3: Check for Cubic and Exponential Growth
A cubic function would show a change in the second differences, which we didn't see here. An exponential function typically grows much faster and would require for the ratios of consecutive \( y \) values to be constant or close to constant, which they're not in this case (the ratios between them are also increasing).
Conclusion
Given that the second differences are constant, the best model for the data is a quadratic function.
Thus, the answer is D: quadratic.