Parameters for Plotting a Cubic Polynomial

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A cubic polynomial is normally characterized by the coefficients However, these coefficients are not well suited to describe the geometrical shape of the graph of the cubic. Nickalls found a set of parameters that do a better job. The relation between the parameter sets is given in the Details section.


The parameters define the plot range and .


Contributed by: Thomas Mueller (March 2011)
Open content licensed under CC BY-NC-SA



The point of inflection N (turning point) has coordinates and . The other two parameters are given by and . The horizontal and vertical distances from N to the minimum are and . But note that Nickalls's parameters and become complex if . The slope at the point of inflection is given by .

A detailed discussion of the parameters is given in:

R. W. D. Nickalls, "A New Approach to Solving the Cubic: Cardan's Solution Revealed," The Mathematical Gazette, 77(480), 1993 pp. 354–359.

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