Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form

(with a ≠ 0) in some Cartesian coordinate system.

Solving for y leads to the explicit form

which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax².)

Solving the implicit equation for x yields a second explicit form

The parametric equation

can also be deduced from the implicit equation by putting ${\textstyle t={\frac {y}{ax}}.}$

The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic.