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Cosine function with a vertical shift: constant, range, sketch, equation and maximum points

Cosine function with a vertical shift: constant, range, sketch, equation and maximum points

IB Year 6 | Grade 12
9 marks

The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=a-2\cos\left(\dfrac{\pi x}{2}\right)\) for \(0\le x\le7\), where \(a\) is a constant. The minimum value of \(\mathrm{f}(x)\) is \(-3\).

  1. Find \(a\) and the range of \(\mathrm{f}\).[2]
  2. On the grid below, sketch the graph of \(y=\mathrm{f}(x)\).[3]
  3. Solve the equation \(\mathrm{f}(x)=-1\).[2]
  4. Find the coordinates of each local maximum point on the graph of \(y=\mathrm{f}(x)\).[2]

Solution:

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Answer:(a) \(a=-1\), range \(-3\le\mathrm{f}(x)\le1\) (c) \(x=1,\ 3,\ 5,\ 7\) (d) \((2,1)\) and \((6,1)\)

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