x	y
−2	15
−1	0.5
0	−6
1	−4.5
2	5
3	22.5
4	48
Find the quadratic function that is the best fit for the given data.
y(x) = 
 

To find the quadratic function that best fits the given data, we can use the method of least squares to find the coefficients of a quadratic function of the form:

\[
y(x) = ax^2 + bx + c
\]

Given the data points:

\[
\begin{align*}
(-2, 15), \\
(-1, 0.5), \\
(0, -6), \\
(1, -4.5), \\
(2, 5), \\
(3, 22.5), \\
(4, 48)
\end{align*}
\]

We'll use these points to set up a system of equations and solve for the coefficients \(a\), \(b\), and \(c\).

Let's calculate the coefficients using the least squares method.The quadratic function that best fits the given data is:

\[
y(x) = 4x^2 - 2.5x - 6
\]

This function has the coefficients:
- \(a = 4\)
- \(b = -2.5\)
- \(c = -6\)

Would you like more details on how these coefficients were calculated, or do you have any other questions?

### Related Questions:
1. How can you interpret the coefficients of a quadratic function?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How do you determine the vertex of the quadratic function \(y(x) = 4x^2 - 2.5x - 6\)?
4. What is the difference between a least-squares fit and an exact fit for a quadratic function?
5. How can you use quadratic regression in real-life applications?

#### Tip:
Quadratic functions can model parabolic relationships, such as the path of a projectile or the curve of a bridge arch.

Find the quadratic function that is the best fit for the given data (-2, 15), (-1, 0.5), (0, -6), (1, -4.5), (2, 5), (3, 22.5), (4, 48).

Learn how to find the quadratic function that best fits a set of data points using the least squares method. Given data includes points (-2, 15), (-1, 0.5), (0, -6), (1, -4.5), (2, 5), (3, 22.5), and (4, 48). This problem explores curve fitting and solving for quadratic function coefficients.

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