Question

Which of the following best models the data in the table?
Type of Function
The table presents values of x and y, showing how y increases as x increases
x y
2 8.5
4 46
6 107.5
8 193
A: cubic B: linear C: exponential D: quadratic

Answer 1

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**.