2024 NJC Promo Q4

The diagram shows the curve \(y=\mathrm{f}\left( x \right)\) that passes through the origin and \(\left( 1,0 \right)\). It has a minimum point \(\left( \frac{1}{2},-\frac{1}{2} \right)\) and asymptotes \(x=-1\) and \(y=3\).

Sketch the curve \(y=\frac{1}{\mathrm{f}\left( x \right)}\), showing clearly the coordinates of any stationary points, axial intercepts and equations of all asymptotes.[3]

A curve with equation \(y=\mathrm{h}\left( x \right)\) undergoes the following sequence of transformations.

Step \(1\): A translation of \(2\) units in the positive direction of the \(x\)-axis.

Step \(2\): A scaling parallel to the \(x\)-axis by factor of \(3\).

Step \(3\): A reflection in the \(y\)-axis.

The resultant curve has equation \(y=-\frac{{{x}^{2}}}{6x+45}\). Find \(\mathrm{h}\left( x \right)\).[3]