A2 June 2021 Paper 1 Q13
13 The transformation S is represented by the matrix \(\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\)
The transformation T is a translation by the vector \(\begin{bmatrix} 0 \\ -5 \end{bmatrix}\)
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]
| Scheme | Marks | AO |
|---|---|---|
| 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 | B1 | 1.2 |
| 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 \(y\) | B1 | 2.4 |
| Completes a rigorous argument to show that Kamla is correct | R1 | 2.1 |
| (3 marks) |
Typical solution
S then T
\[\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3x \\ y \end{bmatrix}\]\[\begin{bmatrix} 3x \\ y \end{bmatrix} + \begin{bmatrix} 0 \\ -5 \end{bmatrix} = \begin{bmatrix} 3x \\ y - 5 \end{bmatrix}\]T then S
\[\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 0 \\ -5 \end{bmatrix} = \begin{bmatrix} x \\ y - 5 \end{bmatrix}\]\[\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} x \\ y - 5 \end{bmatrix} = \begin{bmatrix} 3x \\ y - 5 \end{bmatrix}\]These are the same
So Kamla is correct