Parameters for Plotting a Quartic

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The general quartic can be brought into the reduced form

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by means of the translation . If , then and .

The coordinates of the two points of inflection of are , where .

When , there are two real points of inflection and hence three real turning points. When , both points of inflection are complex and hence there is only one real turning point.

Since , , and are directly related to the geometry of the quartic, this Demonstration offers a more intuitive insight regarding how the shape of the curve is related to the coefficients of the reduced form .

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Contributed by: Thomas Mueller and R.W.D. Nickalls (March 2011)
Open content licensed under CC BY-NC-SA


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The four roots of the reduced quartic equation can be expressed in terms of just three parameters, say , , , where , , are the roots of the resolvent cubic equation , known as Euler's cubic. Note that are the six roots of the resolvent sextic Thus, the key to solving the quartic is to first solve the resolvent cubic

For a more detailed discussion see R. W. D. Nickalls, "The Quartic Equation: Invariants and Euler's Solution Revealed," The Mathematical Gazette, 94, 2009 pp. 66–75.



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