9 The graph shows the function \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\).
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- Describe the transformation of the graph of \(y = e ^ { x }\) that gives the graph of \(y = e ^ { 2 x }\).
A second function is defined by \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\).
- A copy of the graph of \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\) is given in the Printed Answer Booklet.
Add a sketch of the graph of \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\) in a case where \(k\) is a positive constant.
- Show that the two graphs do not intersect for values of \(k\) less than \(- \frac { 1 } { 4 }\).
- In the case where \(k = 2\), show that the only point of intersection occurs when \(x = \ln 2\).