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]