Show that $f(x)$ can be expressed as $1 + \sin 2x$ .

The graph of f is shown below for $0 \le x \le 2\pi$ .

Let $g(x) = 1 + \cos x$ . On the same set of axes, sketch the graph of g for $0 \le x \le 2\pi$ .

The graph of g can be obtained from the graph of f under a horizontal stretch of scale factor p followed by a translation by the vector $\left( {\begin{array}{*{20}{c}} k\\ 0 \end{array}} \right)$ .

Write down the value of p and a possible value of k .

attempt to expand     (M1)

e.g. $(\sin x + \cos x)(\sin x + \cos x)$ ; at least 3 terms

correct expansion     A1

e.g. ${\sin ^2}x + 2\sin x\cos x + {\cos ^2}x$

$f(x) = 1 + \sin 2x$     AG     N0

[2 marks]

     A1A1     N2

Note: Award A1 for correct sinusoidal shape with period $2\pi$ and range $[0{\text{, }}2]$, A1 for minimum in circle.

$p = 2$ , $k = - \frac{\pi }{2}$     A1A1     N2

[2 marks]

Simplifying a trigonometric expression and applying identities was generally well answered in part (a), although some candidates were certainly helped by the fact that it was a "show that" question.

More candidates had difficulty with part (b) with many assuming the first graph was $1 + \sin (x)$ and hence sketching a horizontal translation of $\pi /2$ for the graph of g; some attempts were not even sinusoidal. While some candidates found the stretch factor p correctly or from follow-through on their own graph, very few successfully found the value and direction for the translation.

Part (c) certainly served as a discriminator between the grade 6 and 7 candidates.