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Is there any way, if given multiple points and you wanted to find the equation of the graph where these points lie, how would you find out: First off equation is it? Line, Parabola, Hyperbola, etc? Second, how do you find the equation of this graph?

For example if we are given the points (1,1) (2,4) (3,9) (4,16) and (5,25) how do you find the equation of this graph?

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You can use polynomial interpolation on your points for starters. –  J. M. Jan 26 '12 at 4:36
    
Your question is somewhat similar to this: math.stackexchange.com/questions/85790/… –  Emmad Kareem Jan 26 '12 at 6:21

2 Answers 2

This problem is related to interpolation. One of the big problems is that there are infinitely many functions that go through a set of points, even the points (which seem to 'obviously' lie on a parabola) that you gave us.

One of the most naive types of interpolation is to use Lagrangian polynomial interpolation (mentioned in the link above). It would give a quartic polynomial that goes through those points above. But that is not at all unique.

A bigger problem is that often, it's not clear what makes a certain interpolation any 'simpler' than another. This is a very extensive area of study. For more, look at the polynomial interpolation page.

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Assuming you want a conic section (as implied by your "Line, Parabola, Hyperbola etc"): in general $a x^2 + b x y + c y^2 + d x + e y + f = 0$; you get five linear equations in the parameters $a,b,\ldots f$ by plugging in your given points for $(x,y)$. Solving this system of equations gives $a=-e,\ b = c = d = f = 0$, $e$ arbitrary, and thus of course the equation in this case is $y = x^2$. Five points in "general position" will determine a conic in this way.

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