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Date	May 2022	Marks available	5	Reference code	22M.1.SL.TZ1.6
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	6	Adapted from	N/A

Question

Consider $f (x) = 4 \sin x + 2.5$ and $g (x) = 4 \sin (x - \frac{3 π}{2}) + 2.5 + q$ , where $x \in ℝ$ and $q > 0$ .

The graph of $g$ is obtained by two transformations of the graph of $f$ .

Describe these two transformations.

[2]

The $y$ -intercept of the graph of $g$ is at $(0, r)$ .

Given that $g (x) \geq 7$ , find the smallest value of $r$ .

[5]

Markscheme

translation (shift) by $\frac{3 π}{2}$ to the right/positive horizontal direction A1

translation (shift) by $q$ upwards/positive vertical direction A1

Note: accept translation by $(\begin{matrix} \frac{3 π}{2} \\ q \end{matrix})$

Do not accept ‘move’ for translation/shift.

[2 marks]

METHOD 1

minimum of $4 \sin (x - \frac{3 π}{2})$ is $- 4$ (may be seen in sketch) (M1)

$- 4 + 2.5 + q \geq 7$

$q \geq 8.5$ (accept $q = 8.5$ ) A1

substituting $x = 0$ and their $q (= 8.5)$ to find $r$ (M1)

$(r =) 4 \sin (\frac{- 3 π}{2}) + 2.5 + 8.5$

$4 + 2.5 + 8.5$ (A1)

smallest value of $r$ is $15$ A1

METHOD 2

substituting $x = 0$ to find an expression (for $r$ ) in terms of $q$ (M1)

$(g (0) = r =) 4 \sin (\frac{- 3 π}{2}) + 2.5 + q$

$(r =) 6.5 + q$ A1

minimum of $4 \sin (x - \frac{3 π}{2})$ is $- 4$ (M1)

$- 4 + 2.5 + q \geq 7$

$- 4 + 2.5 + (r - 6.5) \geq 7$ (accept $=$ ) (A1)

smallest value of $r$ is $15$ A1

METHOD 3

$4 \sin (x - \frac{3 π}{2}) + 2.5 + q = 4 \cos x + 2.5 + q$ A1

$y$ -intercept of $4 \cos x + 2.5 + q$ is a maximum (M1)

amplitude of $g (x)$ is $4$ (A1)

attempt to find least maximum (M1)

$r = 2 \times 4 + 7$

smallest value of $r$ is $15$ A1

[5 marks]

Examiners report

Candidates knew aspects of the transformations performed but some were unable to correctly describe them fully, e.g., omitting direction (right/up/positive) or using 'move' instead of translate/shift. Each description requires three parts: transformation type, size and direction. e.g., translation of q units up. For part (b) few candidates were able to fully navigate the reasoning required in this question. A common error was to evaluate $\sin (\frac{- 3 π}{2}) = - 1$ , instead of 1. Those who used sketches to assist in their thinking were typically more successful.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations

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Topic 3— Geometry and trigonometry

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