Standard +0.8 This question requires understanding that matrix transformations and translations don't generally commute, then proving this through either a counterexample or algebraic reasoning. While the concepts are A-level standard, students must recognize that S scales x-coordinates (affecting horizontal position) so the order matters when combined with a vertical translation—a subtle insight that goes beyond routine application of transformations.
The transformation S is represented by the matrix \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\)
The transformation T is a translation by the vector \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\)
Kamla transforms the graphs of various functions by applying first S, then T.
Leo says that, for some graphs, Kamla would get a different result if she applied first T, then S.
Kamla disagrees.
State who is correct.
Fully justify your answer.
[3 marks]
Question 13:
13 | Finds the image of the general
point for one order of application
of S and T
or
Recalls that the matrix for S
represents a stretch parallel to
the x-axis | 1.2 | B1 | S then T
3 0 𝑥𝑥 3𝑥𝑥
� �� � = � �
0 1 𝑦𝑦 𝑦𝑦
3𝑥𝑥 0 3𝑥𝑥
� �+� � = � �
𝑦𝑦 −5 𝑦𝑦−5
T then S
𝑥𝑥 0 𝑥𝑥
� �+� � = � �
𝑦𝑦 −5 𝑦𝑦−5
3 0 𝑥𝑥 3𝑥𝑥
� �� � = � �
These ar0e th1e s𝑦𝑦a−me5 𝑦𝑦−5
So Kamla is correct
Finds the image of the general
point for the alternative order of
application of S and T
or
Explains that S only affects x or
T only affects y | 2.4 | B1
Completes a rigorous argument
to show that Kamla is correct | 2.1 | R1
Total | 3
Q | Marking Instructions | AO | Marks | Typical solution
The transformation S is represented by the matrix $\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$
The transformation T is a translation by the vector $\begin{pmatrix} 0 \\ -5 \end{pmatrix}$
Kamla transforms the graphs of various functions by applying first S, then T.
Leo says that, for some graphs, Kamla would get a different result if she applied first T, then S.
Kamla disagrees.
State who is correct.
Fully justify your answer.
[3 marks]
\hfill \mbox{\textit{AQA Further Paper 1 2021 Q13 [3]}}