I need to fit an expression of the form $f(x,y)$ for which the data comes from an experiment.
From the experiment data I found the following equations.
$$f(x,800)=0.1079x^2−0.1699x+0.4216$$ $$f(x,1000)=0.1088x^2−0.1241x+0.5134$$ $$f(x,1200)=0.1314x^2−0.1589x+0.6241$$
Now I would like to predict the equation for some $y=1300$ or $600$? Is there any known procedure to find $f(x,y)$.
You only gave three equations but it looks that each coefficient is a quadratic function of $y$.
Hopin that you have more curves at fixed $y$, I think it could be a good idea to reuse all data points and fit the data as $$f(x,y)=(a_0+a_1y+a_2y^2)+(b_0+b_1y+b_2y^2)x+(c_0+c_1y+c_2y^2)x^2$$ which just implies multilinear regression.
Now the big question is : how good would be the extrapolation outside the range of given $y$'s?