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 that best fits the given points. The points given are:
.
Quadratic regression involves finding the values of , , and 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.Ask a new question for Free
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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
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