Combined transformations and invariance

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 3 marks
Shape $S$ is reflected in the line $y = x$ to give shape $S_1$ . $S_1$ is then reflected in the $x$ -axis to give shape $S_2$ .
(a) Describe fully the single transformation that maps $S$ onto $S_2$ . (3)
Question 2Non-calculator · 5 marks
A point P $(3,1)$ is rotated 90 degrees clockwise about the origin and then translated by $(3,-2)$ .
(a) Find the final image of P. (3)
(b) Find the final image if the order is reversed. (2)
Question 3Non-calculator · 3 marks
A shape is reflected in the line $x = a$ and then its image is reflected in the line $x = b$ .
(a) Prove that the combined transformation is a translation by the vector with components $2(b - a)$ and 0. (3)

Answers and marks

(a) Rotation $90^\circ$ clockwise about $(0, 0)$

M1 Tracking a point through both reflections, reaching $(y, -x)$ or a correct numerical example such as $(1, 2) \to (2, -1)$ .
A1 Rotation, $90^\circ$ clockwise.
B1 Centre $(0, 0)$ .

(a) $(4,-5)$

(b) $(-1,-6)$

(a) $(x, y) \to (2a - x, y) \to (2b - 2a + x, y)$

M1 The first reflection written generally: $x \to 2a - x$ .
M1 The second reflection applied to $2a - x$ .
A1 Reaching $x + 2(b - a)$ with $y$ unchanged and concluding it is a translation.

Combined transformations and invariance

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