2024 CJC Promo Q9

1. Express \(4{{x}^{2}}-8x+4\) in the form of \(A{{\left( x-B \right)}^{2}}\) where \(A\) and \(B\) are constants to be determined.[1]
   Hence or otherwise, state a sequence of transformations that would transform the curve with equation \(y=3{{e}^{{{x}^{2}}}}\) onto the curve with equation \(y=3{{\mathrm{e}}^{4{{x}^{2}}-8x+4}}-10\).[3]
2. 
   The diagram shows the curve \(y=\mathrm{f}\left( x \right)\). The curve has a turning point at \((6,-5)\) and crosses the \(x\)-axis at \((-4,0)\) and \((0,0)\). The lines \(y=-2\) and \(x=-3\) are the asymptotes to the curve.
   On separate diagrams, sketch the graphs of
   1. \(y=3\left| \mathrm{f}\left( x \right) \right|\) ,[2]
   2. \(y=\frac{1}{\mathrm{f}\left( x \right)}\) ,[3]
   3. \(y={{\mathrm{f}}'}\left( x \right)\) ,[3]

Labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable.