2024 HCI Promo Q7

1. Describe a sequence of two transformations that will transform the graph of \(y=\mathrm{f}\left( x \right)\) to the graph of \(y=\mathrm{f}\left( \alpha x+\beta \right)\), where \(\alpha \) and \(\beta \) are positive constants.[2]

Diagram 1 and Diagram 2 show the graphs of \(y=\mathrm{f}\left( x \right)\) and \(y=\mathrm{f}\left( \alpha x+\beta \right)\) respectively. The turning points with coordinates \(\left( 0,0 \right)\) and \(\left( 2,4 \right)\) on \(y=\mathrm{f}\left( x \right)\) correspond to the points with coordinates \(\left( -2,0 \right)\) and \((2,4)\) respectively on \(y=\mathrm{f}\left( \alpha x+\beta \right)\). The asymptotes \(x=1\) and \(y=x+\lambda \) on \(y=\mathrm{f}\left( x \right)\) correspond to the asymptotes \(x=0\) and \(y=\alpha x+\beta +\lambda \) respectively on \(y=\mathrm{f}\left( \alpha x+\beta \right)\).

2. Find the value of \(\alpha \) and of \(\beta \).[2]
3. On a separate diagram, sketch the graph of \(y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)\), giving the coordinates of the points where the graph crosses the axes and the equations of the asymptotes, if any.[3]