I'm a programmer looking for a way to create polynomial equations from a list of x intercepts and local maxima.(adsbygoogle = window.adsbygoogle || []).push({});

For the sake of discussion we can begin with a function of degree 4. The scale and position of the curve is unimportant so for simplicity's sake the curve can always have x intercepts at 1 and -1. That leaves the declaration of one local maximum to create a unique equation. If I want that maximum to be (0,0) then the equation takes the form (x-1)(x+1)(x)(x). Moving the maximum along the x axis is simple. For maximum at (a,0) use the formula (x-1)(x+1)(x-a)(x-a) for any 'a' where 1>a>-1. Creating an equation where the local maximum is on the y axis is also easy where a maximum at (0,-b) can be given by the equation (x-1)(x+1)(x-bi)(x+bi) for any b where 1>b>0

The problems arise when trying to use a maximum not on a zero axis. for example trying to create a local maximum at (0.2,-0.2) using the equation (x-1)(x+1)(x-.2-.2i)(x-.2+.2i) the local maximum is actually at (0.256,-0.190). Not exactly what I was looking for. So what would be a generalized formula which would give me a local maximum at (a,-b)?

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# I Procedurally generated polynomial functions

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