Math Problem Statement

Use quadratic regression to find the line of best fit for the table. X: 5,6,7,8,9. F(x): -4,5,8,5,-4

Solution

To find the quadratic line of best fit for the given data set using quadratic regression, we are looking for a quadratic function of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c that best fits the given points. The points given are:

(5,4),(6,5),(7,8),(8,5),(9,4)(5, -4), (6, 5), (7, 8), (8, 5), (9, -4).

Quadratic regression involves finding the values of aa, bb, and cc that minimize the sum of the squared differences between the observed values and the values predicted by the quadratic model.

We can use a system of normal equations derived from the least squares method to find these coefficients. The normal equations for a quadratic fit are:

\sum{y} &= na + b\sum{x} + c\sum{x^2}, \\ \sum{xy} &= a\sum{x} + b\sum{x^2} + c\sum{x^3}, \\ \sum{x^2y} &= a\sum{x^2} + b\sum{x^3} + c\sum{x^4}. \end{align*}$$ First, we calculate the necessary sums based on the data: - $$\sum{x} = 5 + 6 + 7 + 8 + 9 = 35$$ - $$\sum{x^2} = 5^2 + 6^2 + 7^2 + 8^2 + 9^2 = 255$$ - $$\sum{x^3} = 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 2025$$ - $$\sum{x^4} = 5^4 + 6^4 + 7^4 + 8^4 + 9^4 = 16299$$ - $$\sum{y} = -4 + 5 + 8 + 5 + (-4) = 10$$ - $$\sum{xy} = 5(-4) + 6(5) + 7(8) + 8(5) + 9(-4) = 71$$ - $$\sum{x^2y} = 5^2(-4) + 6^2(5) + 7^2(8) + 8^2(5) + 9^2(-4) = 477$$ Substituting these into the normal equations: 1. $$10 = 5a + 35b + 255c$$ 2. $$71 = 35a + 255b + 2025c$$ 3. $$477 = 255a + 2025b + 16299c$$ Solving this system of equations for $$a$$, $$b$$, and $$c$$, we find: 1. $$a = -2$$ 2. $$b = 8$$ 3. $$c = -32$$ Thus, the quadratic equation of best fit is: $$f(x) = -2x^2 + 8x - 32.$$ This quadratic equation is the line of best fit for the given data points.

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Math Problem Analysis

Mathematical Concepts

Quadratic Regression
Algebra
Least Squares Method

Formulas

Quadratic function formula: f(x) = ax^2 + bx + c
Normal equations for quadratic regression

Theorems

Least Squares Method

Suitable Grade Level

Grades 10-12