x	y
−2	5
−1	−3.5
0	−8
1	−8.5
2	−5
3	2.5
4	14
Find the quadratic function that is the best fit for the given data.

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:
   \[
   A = \begin{bmatrix}
   x_1^2 & x_1 & 1 \\
   x_2^2 & x_2 & 1 \\
   \vdots & \vdots & \vdots \\
   x_n^2 & x_n & 1 \\
   \end{bmatrix}, \quad
   \mathbf{x} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, \quad
   \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}
   \]

   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.

Find the quadratic function that is the best fit for the given data (-2, 5), (-1, -3.5), (0, -8), (1, -8.5), (2, -5), (3, 2.5), (4, 14).

Learn how to find the best-fit quadratic function for the data points (-2, 5), (-1, -3.5), (0, -8), (1, -8.5), (2, -5), (3, 2.5), and (4, 14) using the least squares method. The solution involves setting up and solving a system of equations to derive the quadratic function y = 2x^2 - 2.5x - 8. This concept is crucial for students studying algebra and data modeling in Grades 10-12.

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