Currently I am working on an assignment for which I have to calculate the quadratic regression and linear regression (I know how to do this one) of some data points by hand. Nonetheless, I do not know how to find the quadratic regression of my data points because I cannot find a correct formula. Additionally, can someone explain how does the quadratic regression work. These are the points.
So, you have $n$ data points $(x_i,y_i)$ and your model is $$y=a+bx+c x^2$$ So, the sum of squares is $$SSQ=\sum_{i=1}^n(a+b x_i+c x_i^2-y_ i)^2$$ As usual, compute the derivatives of SSQ with respect to $(a,b,c)$ and set them equal to $0$. This would give you, just as for the linear case, the so-called normal equations.
In a symbolic form to avoid typing all the sums, they are $$S_y=n a+b S_x+c S_{xx}$$ $$S_{xy}=S_xa+bS_{xx}+cS_{xxx}$$ $$S_{xxy}=S_{xx}a+bS_{xxx}+cS_{xxxx}$$ So, three linear equations for the three unknown variables $(a,b,c)$.
For sure, we could write down the explicit formulae but they would be quite messy