8 The sketch shows the graph of \(y = \cos ^ { - 1 } x\).
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- Write down the coordinates of \(P\) and \(Q\), the end points of the graph.
- Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos ^ { - 1 } x\) onto the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
- Sketch the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
- Write the equation \(y = 2 \cos ^ { - 1 } ( x - 1 )\) in the form \(x = \mathrm { f } ( y )\).
- Hence find the value of \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(y = 2\).