I have this exercise:
Use the Normal Equations to determine the parabola $x^2+bx+c$ that best approximates $(-1,2),(1,-1),(2,1)$ using the least-squares method.
I already solved this. What is the question had been to approximate $ax^2+bx+c$ instead?
I tried to do the same thing, and I used these matrices:
$ \begin{cases} a - b + c = 2 \\ a + b + c = -1 \\4a + 2b + c = 1\end{cases} $
$$A = \begin{bmatrix} 1 & -1 & 1\\ 1 & 1 & 1 \\ 4 & 2 & 1\end{bmatrix} \\ x = \begin{bmatrix} a \\ b \\ c\end{bmatrix}\\ b = \begin{bmatrix} 2 \\ 2 \\ 1\end{bmatrix}$$
I solved for $A^tAx = A^t b$ and got $(-1/3)x^2 + 2.3333333$. I tried this on this website and got these results
I assume a different and more precise method was used. Can someone explain to me why the least-squares method isn't so good for polynomials of degree >1 (assuming I did this right)?
Nevermind this, I made a mistake on b. I could get the same result as the website when I fixed it. Closing.