A-Level Mathematics May/June 2024 Q2(a): In order to transform the curve y = sinx to the curve y = f(x), the curve y = sinx is f…

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  The curve $y = \sin x$ is first reflected in the x-axis to give the curve $y = -\sin x$. The question requires two further transformations to obtain the curve $y = f(x) = -3\sin x - 2$.

  To transform $y = -\sin x$ to $y = -3\sin x - 2$, the following sequence of two transformations is required:

  1. Stretch with a scale factor of 3 parallel to the y-axis. This transforms the curve from $y = -\sin x$ to $y = -3\sin x$.

  2. Translation by the vector $\begin{pmatrix} 0 \\ -2 \end{pmatrix}$. This transforms the curve from $y = -3\sin x$ to $y = -3\sin x - 2$.

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  How the marks are awarded

  • B2 — Awarded for correctly identifying the stretch. Full marks (B2) require all three components: 'Stretch', 'factor 3', and 'in the y-direction' (or 'parallel to the y-axis'). Award B1 if only two of these three components are stated.
  • B2 — Awarded for correctly identifying the translation. Full marks (B2) require both components: 'Translation' (or 'shift') and the vector $\begin{pmatrix} 0 \\ -2 \end{pmatrix}$ (or '2 units down'/'in the negative y-direction'). Award B1 if only one component is correct or if the direction is ambiguous.

  Common mistakes

  • Stating the transformations in the wrong order (translation first, then stretch), which would result in the incorrect final equation $y = -3\sin x - 6$.
  • Incorrectly describing the stretch, for example as 'stretch factor 3 in the x-direction' or omitting the direction entirely.
  • Incorrectly describing the translation, for example as a movement of '2 units up' or using the horizontal vector $\begin{pmatrix} -2 \\ 0 \end{pmatrix}$.
  • Describing the initial reflection in the x-axis as one of the two further transformations, when the question states this has already been applied.

  Examiner tip: When describing transformations applied 'outside' the core function (e.g., $a \cdot f(x) + d$), vertical stretches/compressions (multiplication) must be performed before vertical translations (addition/subtraction).

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