Quadratic function: Difference between revisions

Revision as of 10:47, 26 October 2021

Revision as of 16:14, 3 November 2021

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If <math>a>0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the [[ordinate]] of the [[minimum]] point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!</math>. If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.

If <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either a circle or other ellipse or nothing at all. If the ordinate of the [[maximum]] point of the corresponding parabola

<math> y_p = a x^2 + b x + c \,\!</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an [[Empty set|empty]] locus of points.