Sketching the graphs of f(2|x|) and a reciprocal transformation from a given graph
Sketching the graphs of f(2|x|) and a reciprocal transformation from a given graph
8 marks
The graph of \(y=\mathrm{f}(x)\) is shown below. It has the horizontal asymptote \(y=-2\), crosses the \(x\)-axis at \(x=-2\) and \(x=6\), meets the \(y\)-axis at \((0,3)\) and has a local maximum point at \((2,4)\).
Sketch, on separate sets of axes, the graphs of
- \(y=\mathrm{f}(2|x|)\)[3]
- \(y=\dfrac{8}{\mathrm{f}(x)}\)[5]
On each sketch, label the coordinates of any points where the graph meets the axes and of any maximum or minimum points, and give the equations of any asymptotes.
Solution:
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Answer:(i) intercepts \((\pm3,0)\) and \((0,3)\), maxima \((\pm1,4)\), minimum \((0,3)\), asymptote \(y=-2\) (ii) \(y\)-intercept \(\left(0,\frac83\right)\), minimum \((2,2)\), asymptotes \(x=-2\), \(x=6\), \(y=0\) and \(y=-4\)
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