Find the value of \(k\).

Find the minimum value of \(f(x)\).

Let \(g(x) = \sin x\). The graph of g is translated to the graph of \(f\) by the vector \(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\).

Write down the value of \(p\) and of \(q\).

METHOD 1

attempt to substitute both coordinates (in any order) into \(f\)     (M1)

eg     \(f\left( {\frac{\pi }{4}} \right) = 6,{\text{ }}\frac{\pi }{4} = \sin \left( {6 + \frac{\pi }{4}} \right) + k\)

correct working     (A1)

eg     \(\sin \frac{\pi }{2} = 1,{\text{ }}1 + k = 6\)

\(k = 5\)     A1     N2

[3 marks]

METHOD 2

recognizing shift of \(\frac{\pi }{4}\) left means maximum at \(6\)     R1)

recognizing \(k\) is difference of maximum and amplitude     (A1)

eg     \(6 - 1\)

\(k = 5\)     A1     N2

[3 marks] 

evidence of appropriate approach     (M1)

eg     minimum value of \(\sin x\) is \( - 1,{\text{ }} - 1 + k,{\text{ }}f'(x) = 0,{\text{ }}\left( {\frac{{5\pi }}{4},{\text{ }}4} \right)\)

minimum value is \(4\)     A1     N2

[2 marks]

\(p =  - \frac{\pi }{4},{\text{ }}q = 5{\text{     }}\left( {{\text{accept \(\left( \begin{array}{c} - {\textstyle{\pi  \over 4}}\\5\end{array} \right)\)}}} \right)\)     A1A1     N2

[2 marks]