re "Fitting a polynomial to a set of points" - why you "failed to accomplish what I set out to do"

[hdb3]

Polynomials are not the only mathematical forms which generate 'interesting' graphs, and intuitively not the right forms for building the 'hype cycle' 'function'. One reason is that, in the limit (for large absolute value of 'x'), only a single term dominates (the highest order term), and the graph at scale simply resembles the graph of e.g, x^9, or x^42, or whatever is the highest order term. So, the hype cycle, which appears 'flat' at both limits, cannot be a polynomial.

Simple examples of non-polynomial functions are logarimitihc and trignometric, and these and many others can be constructed with various flavours of 'infinite' expressions, e.g. Fourier series, Taylor series. An interesting and possibly slightly relevant reference would be the ''Heaviside step function', which can be approximated by various analytic expressions (see wikipedia!).

So, your problem I think is in the assumption that there exists some simple finite polynomial which resembles the hype cycle even for large values of 'x'. But, polynomials are only a very small subset of graph generating functions, and rarely useful for real world use cases. If your objective is to match only over a small segment of the 'x' axis, then polynomials can work, but only in that case.

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Disclaimer - I am a software engineer too, not a mathematician.
But, your comment submission format is too long winded to think of using...

I hope this helps.