2023 SAJC Promo Q6

The diagram above shows the graph of \(y=\mathrm{f}\left( x \right)\). The curve crosses the \(x\)-axis at \(\left( -3.5,\,\,0 \right)\) and has a minimum point at \(\left( -2,\,\,0 \right)\). The lines \(x=-3\), \(x=0\) and \(y=3\) are asymptotes of the curve.

On separate diagrams, sketch the graphs in (i) and (ii), stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes and of any turning points in exact form, if possible.

The transformations A, B and C are given as follows:

A: \( \qquad \) Reflection about the \(x\)-axis;

B: \( \qquad \) Translation of \(4\) units in the positive \(x\)-direction;

C: \( \qquad \) Scaling parallel to the \(x\)-axis by a factor of \(2\).

A curve undergoes in succession, the transformations A, B and C, and the equation of the resulting curve is \(y=\frac{1}{3}{{\left( x+1 \right)}^{2}}\).

Determine the original equation of the curve.[3]