Read again what Walter has said. There is much to learn for you.
You might also look at some of what is found in this post, which tries to teach people about how to build nonlinear functional forms, from primitive shapes:
But what you should consider are fundamental behaviors in the process you want to model.
Is this function asymptotically linear on a log scale? Then you want to choose a model that is asymptotically linear. What can I say? A polynomial does NOT have that character. Period.
Next, there are some tricks you can use that can work. First, fit each curve, using a basic model that fits that curve shape well.
Then you take the coefficients from each model. Now model those coefficients as a function of other parameters. For example, suppose you had an entire family of linear models? Thus
z = a1*x + b1 % curve 1
z = a2*x + b2 % curve 2
z = a3*x + b3 % curve 3
But assume that some other parameter was varied to create each of those curves? Now you would fit a as a function of y, as well, fit b as a function of y. So you now have a model of the form
z = a(y)*x + b(y)
Of course, this can get more complicated. But you are the one who understands what your data represents, what models would make sense.