Well, the easy answer is to not do this, at least, don't use a polynomial. People inevitably use too high an order polynomial, since that is what gives the "best" fit. Higher orders are always "better" in terms of error. And then what always happens is they have numerical problems. In fact, I've seen many people having serious numerical problems fitting something as low an order as a cubic polynomial to their data.
As well, higher order polynomials, while they fit the data best, tend to do nasty things between the data points.
And, most of the time, when people use polynomials, it is because they do not know of a better solution. After all, you can represent pretty much any function with a Taylor series, right? So a polynomial model must be a good idea. Easy to do, fast, efficient, etc.
If you insist on the use of a polynomial model, then you can evaluate the function using polyval, but I'm not sure what you mean about taking something intuitively. So I have no idea what you are asking there.
Anyway, a far better idea in general is to use my SLM toolbox for curve fits, if you just want a curve that fits your data. You are given a great deal of control over the resulting shape, and it is designed to fit robustly, avoiding all of those numerical problems I alluded to above.
You can download SLM from the file exchange. The only downside to SLM is it will use the optimization toolbox for many fits.
You evaluate the function using slmeval once it has been fit.
There are other tools you can use too. For simple interpolation, interp1 offers a simple set of spline fits.
