Vertex form and sketch of a quadratic, a quadratic inequality, and an inequality for its composite with e to the x
Vertex form and sketch of a quadratic, a quadratic inequality, and an inequality for its composite with e to the x
15 marks
Let \(\mathrm{f}(x)=2(x-1)(x-5)\), for \(x\in\mathbb{R}\).
- Express \(\mathrm{f}(x)\) as \(a(x-h)^2+k\), where \(a\), \(h\) and \(k\) are integers.[4]
- Sketch the curve \(y=\mathrm{f}(x)\), labelling its vertex and each point where it meets the coordinate axes.[4]
- Solve \(\mathrm{f}(x)\leq10\).[4]
A second function is defined by \(\mathrm{g}(x)=e^x\), for \(x\in\mathbb{R}\).
- Write down \((\mathrm{f}\circ\mathrm{g})(x)\) in terms of \(x\).
- Using your answers to parts (c) and (d)(i), solve \((\mathrm{f}\circ\mathrm{g})(x)\leq10\).[3]
Solution:
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Answer:(a) \(\mathrm{f}(x)=2(x-3)^2-8\) (b) vertex \((3,-8)\), intercepts \((0,10)\), \((1,0)\) and \((5,0)\) (c) \(0\leq x\leq6\) (d)(i) \(2\left(e^x-1\right)\left(e^x-5\right)\) (d)(ii) \(x\leq\ln6\)
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