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Quadratic Regression Calculator

Find parabolas of best fit step by step

The calculator will find the quadratic of best fit for the given set of paired data using the least squares method, with steps shown.

Related calculator: Linear Regression Calculator

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Your Input

Find the parabola of best fit for {(1,0),(4,5),(6,2),(7,1),(3,−3)}\left\{\left(1, 0\right), \left(4, 5\right), \left(6, 2\right), \left(7, 1\right), \left(3, -3\right)\right\}.

Solution

The number of observations is n=5n = 5.

Generate the following table:

xxyyxyx yx2x^{2}x2yx^{2} yx3x^{3}x4x^{4}y2y^{2}
1100001100111100
445520201616808064642562562525
66221212363672722162161296129644
771177494949493433432401240111
33−3-3−9-999−27-272727818199
∑\sum2121553030111111174174651651403540353939

The parabola of best fit is y=ax2+bx+cy = a x^{2} + b x + c.

a=(n(∑x2y)−(∑x2)(∑y))(n(∑x2)−(∑x)2)−(n(∑xy)−(∑x)(∑y))(n(∑x3)−(∑x2)(∑x)))(n(∑x4)−(∑x2)2)(n(∑x2)−(∑x)2)−(n(∑x3)−(∑x2)(∑x))2=(5⋅174−(111)⋅(5))⋅(5⋅111−212)−(5⋅30−(21)⋅(5))⋅(5⋅651−(111)⋅(21))(5⋅4035−1112)⋅(5⋅111−212)−(5⋅651−(111)⋅(21))2=−322a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}

b=(n(∑xy)−(∑x)(∑y))(n(∑x4)−(∑x2)2)−(n(∑x2y)−(∑x2)(∑y))(n(∑x3)−(∑x2)(∑x)))(n(∑x4)−(∑x2)2)(n(∑x2)−(∑x)2)−(n(∑x3)−(∑x2)(∑x))2=(5⋅30−(21)⋅(5))⋅(5⋅4035−1112)−(5⋅174−(111)⋅(5))⋅(5⋅651−(111)⋅(21))(5⋅4035−1112)⋅(5⋅111−212)−(5⋅651−(111)⋅(21))2=32b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}

c=(∑y)−b(∑x)−a(∑x2)n=5−(32)⋅(21)−(−322)⋅(111)5=−2511c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}

Thus, the parabola of best fit is y=−3x222+3x2−2511y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}.

Answer

The parabola of best fit is y=−3x222+3x2−2511≈−0.136363636363636x2+1.5x−2.272727272727273.y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}\approx - 0.136363636363636 x^{2} + 1.5 x - 2.272727272727273.A