Cubic curves

The forms cubic curves can take

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Why cubic curves?

While finding a trend in a data series can be quite useful it is perhaps more interesting to use curves other than lines and more flexible to approximate such series. A candidate type of curve that comes to mind are cubics. Mathematically they are also called curves of the third degree because they represent polynomials of the third degree. A question that rises immediately is: "why not use quadratic curves (they are of the second degree)". The answer to that is: quadratic curves are a special case of cubic curves so we lose nothing by considering cubics only. Moreover cubics have two extrema: a minimum and a maximum, while quadratic curves have only one: a maximum or a minimum. Fig. 3 represents such a curve. It can be considered as a value that has been rising in the past, reached a maximum, and then fell down onto a minimum, but caught up afterwards to rise again.

Fig. 3   A typical cubic with rising trend having a maximum at -1 and a minimum at +1

On the other hand cubics, like straight lines, can be used to represent descending values. Fig. 4 represents such a curve. It represents a value that was declining in the past reached a dip, then raised up for a while and after reaching a maximum continued to fall again.

Fig. 4   A cubic with falling trend having a minimum at -1 and a maximum at +1