Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 9 / b / Solution

Real points require $x\ge 0$, and the two branches areThis 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.