# Finding the formula of an nonlinear function

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/… – NoChance Jan 26 '12 at 6:21

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.