8 The diagram shows the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-6_958_1492_372_242} The curve \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) crosses the \(y\)-axis at the point \(A\) and the curves intersect at the point \(B\).

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } - 1\).
2. Write down the coordinates of the point \(A\).
3. 1. Show that the \(x\)-coordinate of the point \(B\) satisfies the equation $$\left( \mathrm { e } ^ { 2 x } \right) ^ { 2 } - 3 \mathrm { e } ^ { 2 x } - 4 = 0$$
   2. Hence find the exact value of the \(x\)-coordinate of the point \(B\).
4. Find the exact value of the area of the shaded region bounded by the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) and the \(y\)-axis.