2024 MI PRU1 Q5

Describe a sequence of \(2\) transformations that transforms the curve \(C\) with equation \(\frac{x^2}{9} + y^2 = 1\) onto the curve \(D\) with equation \((x - 2)^2 + y^2 = 1\).[2]

The diagram shows the curve with equation \(y = \mathrm{f}(x)\). The curve passes through the origin and the point \((2, 0)\) and has a minimum point at \((3, -4)\). The asymptotes of the curve are the lines \(x = 1\), \(y = 0\) and \(y = 2\).

Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\), indicating the equations of any asymptotes and the coordinates of any points where it crosses the axes and of any turning points, where applicable.[3]

The graph of \(y = \mathrm{g}(x)\) is a semi-circle with its centre at the point \((a, 0)\), where \(a < 0\). The diagram shows the graph of \(y = \mathrm{g}(|x|)\), which intersects the axes at the points \((-b, 0)\), \((b, 0)\) and \((0, c)\), where \(a \neq -b\).[3]

Sketch the graph of \(y = \mathrm{g}(x)\), labelling the coordinates of the centre and the radius of the semi-circle, in terms of \(a\) and/or \(b\).