N2010 P1 Q5

The curve with equation \(y=x^3\) is transformed by a translation of 2 units in the positive \(x\)-direction, followed by a stretch with scale factor \(\frac12\) parallel to the \(y\)-axis, followed by a translation of 6 units in the negative \(y\)-direction.

1. Find the equation of the new curve in the form \(y=\mathrm f(x)\) and the exact coordinates of the points where this curve crosses the \(x\)- and \(y\)-axes. Sketch the new curve.[5]
2. On the same diagram, sketch the graph of \(y=\mathrm f^{-1}(x)\), stating the exact coordinates of the points where the graph crosses the \(x\)- and \(y\)-axes.[3]