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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

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.

Contributed by: Thomas Mueller

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Parameters for Plotting a Cubic Polynomial