AQA Level 2 Further Maths · Level 2 Certificate · Chapter 9: Matrices · Lesson 3

Combining transformations

About 80 minutesNo calculator70 marks

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Date

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The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
$A = (13 - 2 0)$ and $B = (2 - 1 14)$ . Work out $BA$ .
[2 marks]
Answer
A2
Last lesson
Describe fully the single transformation given by the matrix $(0 - 1 10)$ .
[2 marks]
Answer $\dots\dots\dots\dots$
A3
Last lesson
The matrix $(1325)$ maps the point $P$ to the point $(4, 9)$ . Work out the coordinates of $P$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A4
From chapter 8
Work out the equation of the tangent to the curve $y = x^{2} - 4 x + 1$ at the point where $x = 3$ .
[2 marks]
Answer $y = \dots\dots$

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

The point $S (2, - 1)$ is transformed by $P = (3 - 1 12)$ , followed by $Q = (12 - 2 0)$ . Work out the matrix for the combined transformation, and the image of $S$ .

Hint The transformation done first goes on the right: the combined matrix is

QP

, not

PQ

B1
The point $(- 3, 2)$ is transformed by $A = (21 0 - 1)$ , followed by $B = (- 1 2 31)$ . Work out the matrix for the combined transformation, and the image of the point.
[3 marks]
Answermatrix $= \dots\dots$ image $= (\dots\dots, \dots\dots)$

Example 2

A rotation of $9 0^{\circ}$ clockwise about the origin is followed by a reflection in the line $y = - x$ . Work out the matrix for the combined transformation, and describe it as a single transformation.

Hint Write down the two matrices. The reflection is done second, so it goes on the left.

B2
A reflection in the $x$ -axis is followed by a rotation of $27 0^{\circ}$ anticlockwise about the origin. Work out the matrix for the combined transformation, and describe it as a single transformation.
[4 marks]
Answermatrix $= \dots\dots$ $\dots\dots\dots$

Example 3

$A$ is a reflection in the line $y = x$ and $B$ is a rotation of $9 0^{\circ}$ anticlockwise about the origin. Work out $AB$ and $BA$ , and describe each.

Hint

BA

means

A

first, then

B

. Work out both products and compare.

B3
$G = (0 - 1 10)$ and $H = (- 2 0 0 - 2)$ . Work out $GH$ and $HG$ . What do you notice?
[3 marks]
Answer $GH = \dots\dots HG = \dots\dots$

Example 4

$R = (01 - 1 0)$ . Work out $R^{2}$ and say what it means as a transformation. Hence write down $R^{4}$ .

Hint

R^{2} = RR

R

is a quarter turn: what are two quarter turns?

B4
$S = (0 - 1 - 1 0)$ . Describe the transformation $S$ represents. Work out $S^{2}$ , and explain your answer using transformations.
[3 marks]
Answer $S^{2} = \dots\dots$

Example 5

A point $Q$ is transformed by $A = (1021)$ , then by $B = (01 - 1 0)$ , then by $C = (3003)$ . Its final image is $(6, 3)$ . Work out the coordinates of $Q$ .

Hint The combined matrix is

CBA

: last on the left. Work out

BA

first (or

CB

first, it gives the same), then solve.

B5
A point $P$ is transformed by $A = (1201)$ , then by $B = (- 1 0 01)$ , then by $C = (0110)$ . Its final image is $(4, 1)$ . Work out the coordinates of $P$ .
[4 marks]
Answer $P = (\dots\dots, \dots\dots)$

Practice

Level 2

C1
A rotation of $18 0^{\circ}$ about the origin is followed by a reflection in the $y$ -axis. Work out the matrix for the combined transformation, and describe it as a single transformation.
[3 marks]
Answermatrix $= \dots\dots$ $\dots\dots\dots$
C2
Use matrices to show that an enlargement, scale factor $- 4$ , centre the origin, followed by an enlargement, scale factor $\frac{1}{2}$ , centre the origin, is a single enlargement, centre the origin, with scale factor $k$ . Write down the value of $k$ .
[3 marks]
Answer $k = \dots\dots$

C3
A rotation of $9 0^{\circ}$ anticlockwise about the origin is followed by a transformation $T$ . Every point ends where it started. Write down the matrix for $T$ and describe $T$ .
[2 marks]
Answer $T = \dots\dots$
C4
$P = (21 a 3)$ and $Q = (b 0 1 - 1)$ . The transformation $P$ followed by the transformation $Q$ has matrix $(7 - 1 - 3 - 3)$ . Work out the values of $a$ and $b$ .
[3 marks]
Answer $a = \dots\dots b = \dots\dots$
C5
The point $(4, - 7)$ is reflected in the line $y = - x$ and then rotated $9 0^{\circ}$ anticlockwise about the origin. Use a single matrix to work out its final image.
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Exam-style questions

AQA exam style

D1
Transformation $A$ is a reflection in the $y$ -axis. Transformation $B$ is a reflection in the $x$ -axis. Which matrix represents $A$ followed by $B$ ? Circle your answer.
$(10 0 - 1) (- 1 0 01) (- 1 0 0 - 1) (0 - 1 - 1 0)$
[1 mark]

D2
Ravi says, "A rotation of $9 0^{\circ}$ clockwise about the origin followed by a reflection in the $y$ -axis gives the same as the reflection followed by the rotation. You multiply the same two matrices."
Use matrices to show that Ravi is wrong. Describe each of the two combined transformations.
[4 marks]
D3
$A$ , $B$ and $C$ are transformations in the $x$ - $y$ plane.
$A$ : a rotation of $18 0^{\circ}$ about the origin
$B$ : a reflection in the line $y = - x$
$C$ : transformation $A$ followed by transformation $B$
Use matrix multiplication to show that $C$ is equivalent to a single reflection.
[4 marks]
D4
$M = (10 k 2)$ and $N = (21 0 - 1)$ , where $k$ is a constant.
(a)
Work out the matrix for $M$ followed by $N$ . Give your answer in terms of $k$ .
[2 marks]
Answer $\dots\dots\dots$
(b)
$M$ followed by $N$ maps the point $(3, 1)$ to the point $(14, 5)$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
(c)
Hence work out the point that $M$ followed by $N$ maps to $(12, 4)$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D4: 7 marks
D5
Triangle $T$ is shown on the grid. $T$ is rotated $9 0^{\circ}$ clockwise about the origin to give triangle $U$ . $U$ is then reflected in the $x$ -axis to give triangle $V$ .
(a)
Work out the matrix that maps $T$ to $V$ .
[3 marks]
Answer $\dots\dots\dots$
(b)
Describe fully the single transformation that maps $T$ to $V$ .
[1 mark]
Answer $\dots\dots\dots\dots$
(c)
Hence write down the coordinates of the vertices of $V$ .
[3 marks]
Answer $\dots\dots\dots$
Total for D5: 7 marks
D6
Matrix $P$ represents a reflection in the $x$ -axis. Matrix $Q$ represents an enlargement, scale factor $4$ , centre the origin. The point $A$ is transformed by $P$ followed by $Q$ to the point $(- 8, 12)$ . Work out the coordinates of $A$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Transformation $A$ is a reflection in the $x$ -axis. Transformation $B$ is a rotation of $9 0^{\circ}$ anticlockwise about the origin. A point $P$ is transformed by $A$ , then $B$ , then $A$ , then $B$ , and so on, taking turns: $2027$ transformations in all, starting and ending with $A$ . The final image is $(5, - 2)$ . Work out the coordinates of $P$ .
[5 marks]
Hint 1 · What to try
Work out the matrix for $A$ followed by $B$ . What does that pair do if it is done twice?
Hint 2 · The first line
$BA = (0110)$ , a reflection in $y = x$ , so two pairs do nothing. $2027 = 2 \times 1013 + 1$ .
Hint 3 · The full method
There are $1013$ pairs and one more $A$ . $1013$ is odd, so the pairs together are one $BA$ . The whole chain is $A (BA) = (0 - 1 10)$ . Solve $(0 - 1 10) (x y) = (5 - 2)$ : $y = 5$ and $x = 2$ , so $P$ is $(2, 5)$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the single matrix for one transformation followed by another, multiplying in the right order.
I can describe a combined transformation as one single transformation.
I can show when the order of two transformations matters, and interpret a power such as $R^{2}$ .
I can work back from an image to the point it came from under a combined transformation.

Answers and mark scheme

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