I have following 4 graphs and for each I have to do regression.
The relation is clearly curvilinear. What term should I use for regression (eg y ~ x+x^2) for each of these?
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I extracted a few points from curve A and fairly successfully fitted a quartic (degree 4) polynomial. Lower order didn't fit very well. I suspect it should fit the other curves equally well.
I suggest you try your regression with $x$, $x^2$, $x^3$ and $x^4$ terms, but bear in mind that the fit is only for the range you fit - it probably shouldn't be used for any extrapolation without an understanding of the physical processes (if any) underlying.