Unique polynomials

You have an original functional shape, it could be a rather complex one. You then approximate it with a polynomial of degree 1 googol.

Then you select a tiny part of the curve of that polynomial. It is so tiny that it looks just like a small straight line.

Is it then true, that that little part contains the information that defines the entire shape? In other words, that this part contains the information of every term in the polynomial, i.e. the coefficient and the power degree of every term?

This would then also then imply that no other polynomial has that little part in its curve. Other polynomials might have parts that look extremely similar, but not exactly the same.