Combined transformations

Higher only

The idea

Do the first transformation, then the second on the result. The order matters. An invariant point stays in the same place after the transformation.

Draw the image after the first transformation.
Transform that image with the second one.
Describe the single transformation from the original to the final shape, if asked.

Watch out Reflect then rotate does not usually give the same result as rotate then reflect.

Worked exampleNon-calculator3 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 $S_2$ .

Describe fully the single transformation that maps $S$ onto $S_2$ .

Follow a general point: $(x, y) \to (y, x)$ after the first reflection.
Then $(y, x) \to (y, -x)$ after reflecting in the $x$ -axis.
$(x, y) \to (y, -x)$ is a rotation of $90^\circ$ clockwise about the origin. Check: $(1, 0) \to (0, -1)$ .

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

Question 1Non-calculator3 marks

Point P( $-2$ , 4) is reflected in the line x = 1. Its image is then reflected in the line x = 5. Work out the coordinates of the final image.

Write your answer, then show the working

Answer (6, 4)

Question 2Non-calculator5 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.

(b) Find the final image if the order is reversed.

Question 3Non-calculator3 marks

A shape is reflected in the line $x = a$ and then its image is reflected in the line $x = b$ .

Prove that the combined transformation is a translation by the vector with components $2(b - a)$ and 0.

Write your answer, then show the working

Reflecting in $x = a$ keeps $y$ and sends $x$ to the same distance on the other side: $x \to 2a - x$ .
Reflecting that in $x = b$ : $2a - x \to 2b - (2a - x) = x + 2(b - a)$ .
So $(x, y) \to (x + 2(b - a),\; y)$ for every point: a translation by $2(b - a)$ across and 0 up.

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

25 official questions in 25 papers since 2017. The latest:

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