To find the quadratic function that best fits the given data, we use the general form of a quadratic equation:

$y = ax^2 + bx + c$

The goal is to find the values of $a$, $b$, and $c$ that minimize the difference between the actual $y$-values and the $y$-values predicted by the quadratic equation.

The given data points are:

$(-2, 5), (-1, -3.5), (0, -8), (1, -8.5), (2, -5), (3, 2.5), (4, 14)$

Steps to calculate the best-fit quadratic function:

1. Set up the system of equations by plugging in the data points into the quadratic equation. We use the least-squares method to minimize the sum of the squared residuals (differences between observed and predicted $y$-values).

2. Construct matrices and solve the system. We can write this in matrix form $A \mathbf{x} = \mathbf{y}$, where:

   x_1^2 & x_1 & 1 \\ x_2^2 & x_2 & 1 \\ \vdots & \vdots & \vdots \\ x_n^2 & x_n & 1 \\ \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, \quad \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$$ This allows us to solve for the values of $$a$$, $$b$$, and $$c$$ that best fit the data.

Let me calculate the values for $a$, $b$, and $c$.The quadratic function that best fits the given data is:

$y = 2x^2 - 2.5x - 8$

This function minimizes the difference between the actual $y$-values and the predicted $y$-values based on the quadratic model.

Would you like more details on how this solution was derived, or have any further questions?

Here are 5 related questions to expand on this topic:

1. How is the least squares method used to fit data to a quadratic model?
2. How can this quadratic function be used to make predictions for new $x$-values?
3. What is the significance of each coefficient $a$, $b$, and $c$ in the quadratic function?
4. How would the best-fit model change if additional data points were added?
5. What are the differences between fitting linear and quadratic functions to data?

Tip: Quadratic functions are useful for modeling data that exhibits curvature, unlike linear functions which assume a straight-line relationship between variables.