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