Find the coordinates of the points \(P , Q\) and \(R\).
Sketch, on separate diagrams, the graphs of
\(y = \mathrm { f } ( 2 x )\),
\(y = \mathrm { f } ( | x | + 1 )\),
indicating on each sketch the coordinates of any maximum points and the intersections with the \(x\)-axis.
(6)
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Figure 2 shows a sketch of part of the curve \(C\), with equation \(y = \mathrm { f } ( x - v ) + w\), where \(v\) and \(w\) are constants. The \(x\)-axis is a tangent to \(C\) at the minimum point \(T\), and \(C\) intersects the \(y\)-axis at \(S\). The line joining \(S\) to the maximum point \(U\) is parallel to the \(x\)-axis.
Find the value of \(v\) and the value of \(w\) and hence find the roots of the equation
$$f ( x - v ) + w = 0$$