Plot-intuition, remarkable or just an algorithm I find it rather amazing that by plotting the points (0,0),(0.5,0.25),(1,1),(2,4), one can "predict" what the graph will look like. In certain cases, a person may even be able to "sketch"(freehand) the in between values by "connecting the dots", without being told that the function giving the result is $f(x)=x^2$. 
Is there any way to get from a set of data to a prediction of what the plot will look like by having just a few data points?
 A: "In engineering and science one often has a number of data points, as obtained by sampling or experimentation, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points."
Actually this is a very difficult task and there are quite a few approaches for it, the easiest one is probably that you assume the function has a certain form like $f(x)=a\exp(b\cdot x)+c$ and then optimize the parameters a,b and c to get the "best fitting" function. Of course the initial guess is very important and can also be automated. Like already mentioned when working in a very abstract context you are never sure if some very odd / oscilating function is hiding between the data points, but one might assume that this is not the case if you have enough data points.
This is a very neat example from Wikipedia that shows that there are also interpolations which you wouldn't probably come up when fitting it from hand but if you optimize it by using splines on a formula level:

Image and quote: http://en.wikipedia.org/wiki/Interpolation
A: This is the starting point of a whole theory called interpolation.
