= Solution
Real points require $x\geq0$, and the two branches are
$$
\boxed{y=\pm x^{3/2},\qquad x\geq0.}
$$
This is a <semicubical parabola>, symmetric about the horizontal axis and with a cusp at the origin. The parametrization $x=t^2$, $y=t^3$ traces both branches. Their limiting slopes at the origin are zero, so the common tangent direction is the positive horizontal axis, even though the curve has a cusp there.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1-semicubical-parabola.png]
{title=The two real branches of the semicubical parabola and their cusp at the origin}
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