Question

Which of the following best models the data in the table?

Type of Function

x y
2
8.5

4
46

6
107.5

8
193

(1 point)
Responses

linear
linear

quadratic
quadratic

cubic
cubic

exponential

Answer 1

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:

\[
\begin{align*}
x & : 2, 4, 6, 8 \\
y & : 8.5, 46, 107.5, 193 \\
\end{align*}
\]

1. **Calculate the differences of \( y \):**

\[
\begin{align*}
y(4) - y(2) & : 46 - 8.5 = 37.5 \\
y(6) - y(4) & : 107.5 - 46 = 61.5 \\
y(8) - y(6) & : 193 - 107.5 = 85.5 \\
\end{align*}
\]

2. **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**.