Does anyone know a way to identify a function of a given graph?
Ie. you have the max and min points, values where it crosses the x-axis etc.
No! There is no way.
Assume the graph of a function look like a circle. But how do you know it is a circle? Just because it looks like it, No! Thus, there is not way to state what a function is precisely. Or another example. If it looks like a line does not means it is a line maybe it has a small bend somewhere. We do not know.
However, there is a way to approximate functions. Yes.
1)If you want to approximate a curve through some finite set of points you first look at the shape they form. For example, if it looks like it is an exponential you look for an approximating function of the form $\displaystyle y=A\cdot B^x$. And there are techinques to get them, called Method of Least Squares. Look here. (I explain the most basic case).
2)If you want approximate a curve with another given curve, the set of points is infinite, worse, uncountable! In that case you miminized,
$\displaystyle \int_a ^b (f(x)-A(x))^2 dx$.
Where $\displaystyle A(x)$ the the approximating curve.
Look here. (For an example).