## 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

Comma-separated.
Comma-separated.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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

### Solution

The number of observations is $n = 5$.

Generate the following table:

 $x$ $y$ $x y$ $x^{2}$ $x^{2} y$ $x^{3}$ $x^{4}$ $y^{2}$ $1$ $0$ $0$ $1$ $0$ $1$ $1$ $0$ $4$ $5$ $20$ $16$ $80$ $64$ $256$ $25$ $6$ $2$ $12$ $36$ $72$ $216$ $1296$ $4$ $7$ $1$ $7$ $49$ $49$ $343$ $2401$ $1$ $3$ $-3$ $-9$ $9$ $-27$ $27$ $81$ $9$ $\sum$ $21$ $5$ $30$ $111$ $174$ $651$ $4035$ $39$

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

$a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}$

$b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}$

$c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}$

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

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