Show that $g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4$.

Find the range of $g \circ f$.

Given that $g \circ f\left( {\frac{{3\pi }}{{20}}} \right) = 7$, find the next value of $x$, greater than ${\frac{{3\pi }}{{20}}}$, for which $g \circ f(x) = 7$.

The graph of $y = g \circ f(x)$ can be obtained by applying four transformations to the graph of $y = \sin x$. State what the four transformations represent geometrically and give the order in which they are applied.

$g \circ f(x) = g\left( {f(x)} \right)$     M1

$= g\left( {2x + \frac{\pi }{5}} \right)$

$= 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4$     AG

[1 mark]

since $- 1 \le sin \theta  \le  + 1$, range is $[1,{\text{ }}7]$     (R1)A1

[2 marks]

$3\sin \left( {2x + \frac{\pi }{5}} \right) + 4 = 7 \Rightarrow 2x + \frac{\pi }{5} = \frac{\pi }{2} + 2n\pi  \Rightarrow x = \frac{{3\pi }}{{20}} + n\pi$     (M1)

so next biggest value is $\frac{{23\pi }}{{20}}$     A1

Note:     Allow use of period.

[2 marks]

Note:     Transformations can be in any order but see notes below.

stretch scale factor $3$ parallel to $y$ axis (vertically)     A1

vertical translation of $4$ up     A1

Note:     Vertical translation is $\frac{4}{3}$ up if it occurs before stretch parallel to $y$ axis.

stretch scale factor $\frac{1}{2}$ parallel to $x$ axis (horizontally)     A1

horizontal translation of $\frac{\pi }{{10}}$ to the left     A1

Note:     Horizontal translation is $\frac{\pi }{{5}}$ to the left if it occurs before stretch parallel to $x$ axis.

Note:     Award A1 for magnitude and direction in each case.

Accept any correct terminology provided that the meaning is clear eg shift for translation.

[4 marks]

Total [9 marks]

Well done.

Generally well done, some used more complicated methods rather than considering the range of sine.

Fine if they realised the period was $\pi$, not if they thought it was $2\pi$.

Typically 3 marks were gained. It was the shift in the axis $\chi$ of $\frac{\pi }{{10}}$ that caused the problem.