The diagram below shows the graph of \(y=\dfrac1{\mathrm{f}(x)}\) with stationary points at \(\left(0,\dfrac15\right)\) and \(\left(3,\dfrac12\right)\). The lines \(y=0\) and \(x=-3\) are asymptotes of the graph.
Sketch the following graphs on separate diagrams, showing clearly the coordinates of the axial intercepts, turning points as well as equations of any asymptotes:
\(y=\dfrac{1}{\mathrm{f}(|x+1|)}\),[3]
\(y=\mathrm{f}(x)\).[3]
Describe a pair of transformations which transforms the graph of \(y=2e^{-3x}\) onto the graph of \(y=e^{3x+1}\).[2]
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Answer:(a)(i) Min \(\left(-1,\dfrac15\right)\); maxima \(\left(-4,\dfrac12\right),\left(2,\dfrac12\right)\); asymptote \(y=0\); no x-intercepts; y-intercept \(\left(0,\dfrac1{\mathrm{f}(1)}\right)\). (ii) Max \((0,5)\), min \((3,2)\), intended x-intercept \((-3,0)\). (b) Reflect in the y-axis, then stretch parallel to the y-axis by factor \(\dfrac e2\).