A-Level Mathematics October/November 2024 Q5(a): Describe fully a sequence of three transformations which transforms the graph of y = f(…

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  To transform the graph of $y = f(x)$ to the graph of $y = g(x)$, the following sequence of three transformations can be applied:

  1. Reflection in the y-axis. This is the transformation where every point $(x, y)$ on the graph is mapped to $(-x, y)$. The equation becomes $y = f(-x)$.

  2. Stretch parallel to the y-axis with a scale factor of 2. This transformation maps every point $(x, y)$ to $(x, 2y)$. The equation becomes $y = 2f(-x)$.

  3. Translation by the vector $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$. This transformation maps every point $(x, y)$ to $(x-1, y)$. The equation becomes $y = 2f(-(x+1))$, which simplifies to $y = 2f(-x-1)$.

  Note: The vertical stretch (step 2) can be performed at any point in the sequence. However, the reflection in the y-axis must be performed before the horizontal translation for this specific sequence to be correct.

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  How the marks are awarded

  • B1 — Correctly identifying one of the transformations as a 'Reflection'.
  • B1 — Correctly and fully describing the reflection as being 'in the y-axis'.
  • B1 — Correctly identifying the 'Translation' and describing it fully with the correct vector, $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$.
  • B2 — Correctly identifying all three required transformations with their full descriptions: Reflection in the y-axis, Stretch parallel to the y-axis with scale factor 2, and Translation by vector $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$. (A B1 mark is awarded if any two of these three are fully correct).
  • DB1 — Providing a complete and correct sequence of exactly three transformations. The transformations must be correctly named and the order must be valid. This mark is dependent on the other marks being awarded and is lost if extra transformations are included or if the order is incorrect (e.g. translating horizontally before reflecting).

  Common mistakes

  • Giving an incorrect order for the horizontal transformations, for example stating the translation by vector (-1, 0) occurs before the reflection in the y-axis, which would result in a different final graph.
  • Using imprecise language for the translation, such as 'move one unit to the left', which is not accepted. A vector or a full description like 'a translation of 1 unit in the negative x-direction' is required.
  • Omitting one of the three transformations, most commonly the vertical stretch, which prevents scoring full marks.
  • Confusing the axes of transformation, for example, stating a 'reflection in the x-axis' or a 'stretch parallel to the x-axis'.

  Examiner tip: For questions involving multiple transformations, always write down the sequence step-by-step and consider the order carefully, as horizontal transformations (reflections, stretches, translations) are not always commutative.

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