Sketch the graph of \(y = 3 ^ { x }\), stating the coordinates of the point where the graph crosses the \(y\)-axis.
Describe a single geometrical transformation that maps the graph of \(y = 3 ^ { x }\) :
onto the graph of \(y = 3 ^ { 2 x }\);
onto the graph of \(y = 3 ^ { x + 1 }\).
Using the substitution \(Y = 3 ^ { x }\), show that the equation
$$9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$$
can be written as
$$( Y - 1 ) ( Y - 2 ) = 0$$
Hence show that the equation \(9 ^ { x } - 3 ^ { x + 1 } + 2 = 0\) has a solution \(x = 0\) and, by using logarithms, find the other solution, giving your answer to four decimal places.
(4 marks)