Question

Find the best fit parabola through the points (0,0), (1,2), (4,4), (3,6), (2,3), (10,13)

Answer 1

A solution with comments and plots is attached.

Answer 2

ok so we have a least squares problem.....
first we know that the general form parabola is:
y=ax^2+bx+c
now we plug in the values so we'll get the linear equations:
c=0
a+b+c=2
16a+4b+c=4
9a+3b+c=6
4a+2b+c=3
100a+10b+c=13

this will give us the matrix:
\[\left[\begin{matrix}0 & 0 & 1\\ 1 & 1 & 1\\16 & 4 & 1\\9 & 3 & 1\\4 & 2 & 1\\100 & 10 & 1\end{matrix}\right]\left(\begin{matrix}a \\ b\\ c\end{matrix}\right)=\left(\begin{matrix}0 \\2\\4\\6\\3\\13\end{matrix}\right)\]
lets rewrite this as
Ax=b
where A is the 6x3 matrix, x is a column matrix containing a,b,c and b is the column matrix in the RHS.
now recall that we want to find the values of a and b and c that will give us the best approximation for the parabola that will fit through these points. to do that recall that we are to to solve the following normal system associated with our system of linear equation which is:
\[A^TAx=A^Tb\]
where A^T is the transpose of A. tell me if I need to show you how to solve this system of linear equation @mellylol :D.

Answer 3

Yes plz @anonymoustwo44