How do you find the function of a curved graph based on a set of points?

Given a set of n points it is possible to find a polynomial of order n-1 to fit them.

So if we have 3 points, that means there's a polynomial of order 2 that will pass through all the points. A polynomial of order 2 is also known as a quadratic, or parabola.

Let's use an example to illustrate:

Take the 3 points (0,0), (1,1), (2,0)

Then these will fit a curve with the equation y = ax2+bx+c

If we put the first point into this equation (y=0, and x=0), then that means: 0 = 0+0+c

so c=0.

The second point (y=1, and x=1) shows us that: 1 = a.1+b.1+c = a+b (since c=0).

so 1 = a+b.

The third point says that: 0 = 4a+2b, so b = -2a.

We now have a simultaneous equation in a and b.

(1): a+b=1

(2): b=-2a

These can easily be solved by substitution giving: a=-1, b=2

Therefore we have the curve y = -x2+2x

which fits the three points given above.

Technical note: It can be the case that the polynomial required has an order of less than n-1 for n points, but it will never be more.