Math Problem Statement

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) =

Solution

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)=ax2+bx+cy(x) = ax^2 + bx + c

Given the data points:

(-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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Least Squares Method
Curve Fitting

Formulas

y(x) = ax^2 + bx + c
Least squares method to find coefficients

Theorems

Least Squares Approximation

Suitable Grade Level

Grades 10-12

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