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

Transformations and the unit square

About 80 minutesNo calculator62 marks

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Date

Do Now

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

A1
Needed today
Write down the image of the point $(5, - 2)$ after (a) a rotation of $9 0^{\circ}$ anticlockwise about $O$ and (b) a reflection in the line $y = - x$ .
[2 marks]
Answer
A2
Last lesson
Work out $(23 - 1 4) (1 - 2 50)$ .
[2 marks]
Answer
A3
Last lesson
Given that $(32 k 4) (4 - 2 - 5 3) = 2 I$ , work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
A4
From chapter 8
Work out the $x$ -coordinates of the stationary points of $y = x^{3} - 6 x^{2} + 9 x$ .
[2 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$

Example, then your turn

Level 2

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

Example 1

Work out the image of the point $(- 4, 3)$ under the transformation matrix $(2 - 1 53)$ .

Hint Write the point as a column vector and multiply: the matrix goes first.

B1
Work out the image of the point $(6, - 2)$ under the transformation matrix $(- 3 4 1 - 2)$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

Example 2

The matrix $(k 2 3 - 1)$ maps the point $(4, - 2)$ to the point $(14, 10)$ . Work out the value of $k$ .

Hint Multiply the matrix by the column vector

(4 - 2)

and set the top entry equal to

14

B2
The matrix $(2 q p - 3)$ maps the point $(3, 4)$ to the point $(- 2, 15)$ . Work out the values of $p$ and $q$ .
[3 marks]
Answer $p = \dots\dots q = \dots\dots$

Example 3

The matrix $(3124)$ maps the point $P$ to the point $(9, 13)$ . Work out the coordinates of $P$ .

Hint Call

P

the point

(x, y)

. Multiplying gives two equations in

x

and

y

. Solve them.

B3
The matrix $(25 - 3 1)$ maps the point $Q$ to the point $(- 11, - 2)$ . Work out the coordinates of $Q$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Example 4

Work out the matrix for a rotation of $9 0^{\circ}$ anticlockwise about the origin. Show the image of the unit square $O A B C$ on the grid, labelling each vertex.

Hint Find where

(1, 0)

and

(0, 1)

go. Their images, written as columns side by side, are the matrix.

B4
Work out the matrix for a reflection in the line $y = - x$ . On the grid, draw the image of the unit square $O A B C$ under this reflection and label $A^{'}$ , $B^{'}$ and $C^{'}$ .
[3 marks]
Answermatrix $= \dots\dots\dots$

Example 5

Describe fully the single transformation given by the matrix $(- 3 0 0 - 3)$ . The unit square is transformed by this matrix. Work out the area of the image.

Hint Find the images of

(1, 0)

and

(0, 1)

. Then the lengths of the sides of the image.

B5
The unit square $O A B C$ is transformed by the matrix $(h 0 0 h)$ to $O A^{'} B^{'} C^{'}$ . The area of $O A^{'} B^{'} C^{'}$ is $50$ . Work out the two possible values of $h$ . Give your answers in surd form.
[3 marks]
Answer $h = \dots\dots$ or $h = \dots\dots$

Practice

Level 2

C1
Describe fully the single transformation given by the matrix $(- \frac{1}{2} 0 0 - \frac{1}{2})$ .
[2 marks]
Answer $\dots\dots\dots\dots$
C2
The matrix $(23 - 1 1)$ maps the point $(a, 4)$ to the point $(b, 25)$ . Work out the values of $a$ and $b$ .
[3 marks]
Answer $a = \dots\dots b = \dots\dots$

C3
A transformation maps $(1, 0)$ to $(2, 5)$ and $(0, 1)$ to $(- 1, 3)$ . Write down its matrix. Hence work out the image of the point $(4, - 2)$ .
[3 marks]
Answermatrix $= \dots\dots$ image $= (\dots\dots, \dots\dots)$
C4
Triangle $T$ has vertices $(1, 1)$ , $(3, 1)$ , $(1, 2)$ . It is transformed by $(- 2 0 0 - 2)$ . Work out the vertices of the image of $T$ , and its area.
[4 marks]
Answervertices $\dots\dots\dots$ area $= \dots\dots$
C5
A point $P$ is reflected in the line $y = x$ . Its image is $(- 4, 9)$ . Write down the matrix for the reflection, and use it to work out the coordinates of $P$ .
[2 marks]
Answer $P = (\dots\dots, \dots\dots)$

Exam-style questions

AQA exam style

D1
Which matrix represents a reflection in the line $y = x$ ? Circle your answer.
$(10 0 - 1) (0110) (0 - 1 - 1 0) (- 1 0 01)$
[1 mark]

D2
Ella says, "The matrix $(01 - 1 0)$ represents a rotation of $9 0^{\circ}$ clockwise about the origin."
Show that Ella is wrong. Then write down the matrix for a rotation of $9 0^{\circ}$ clockwise about the origin.
[3 marks]
Answermatrix $= \dots\dots\dots$
D3
The matrix $(13 2 - 1)$ maps the point $(p, q)$ to the point $(p + 8, 2 q)$ . Work out the values of $p$ and $q$ . You must show your working.
[4 marks]
Answer $p = \dots\dots q = \dots\dots$
D4
The line $L$ has equation $y = 2 x - 1$ . Every point of $L$ is transformed by the matrix $(31 - 1 2)$ .
Show that every image point lies on the line $y = 5 x - 7$ .
[4 marks]
Not drawn accurately
D5
The diagram shows the unit square $O A B C$ and its image $O A^{'} B^{'} C^{'}$ under a transformation $T$ .
(a)
Write down the matrix for $T$ .
[2 marks]
Answer $\dots\dots\dots$
(b)
Describe fully the single transformation $T$ .
[2 marks]
Answer $\dots\dots\dots\dots$
(c)
$T$ maps the point $Q$ to the point $(8, - 6)$ . Work out the coordinates of $Q$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D5: 6 marks
D6
$M = (2 k k 2)$ , where $k$ is a constant. $M$ maps the point $(3, - 1)$ to a point on the line $x + y = 10$ . Work out the value of $k$ , and the coordinates of the image point.
[3 marks]
Answer $k = \dots\dots (\dots\dots, \dots\dots)$

Extension

Stretch

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

E1
The unit square $O A B C$ is transformed by the matrix $(34 - 4 3)$ to $O A^{'} B^{'} C^{'}$ . Show that $O A^{'} B^{'} C^{'}$ is a square, and work out its area.
[5 marks]
Hint 1 · What to try
Find $A^{'}$ , $B^{'}$ and $C^{'}$ by multiplying. Then look at the lengths of the sides and the angle at $O$ .
Hint 2 · The first line
$A^{'} = (3, 4)$ , $B^{'} = (- 1, 7)$ and $C^{'} = (- 4, 3)$ . The length $O A^{'}$ is $3^{2} + 4^{2}$ .
Hint 3 · The full method
$O A^{'} = A^{'} B^{'} = B^{'} C^{'} = C^{'} O = 5$ . The gradient of $O A^{'}$ is $\frac{4}{3}$ and of $O C^{'}$ is $- \frac{3}{4}$ ; their product is $- 1$ , so the angle at $O$ is a right angle. A rhombus with a right angle is a square, of area $5^{2} = 25$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the image of a point under a matrix, and the point that maps to a given image.
I can write down the matrix for a reflection, a rotation about the origin or an enlargement centre the origin.
I can draw the image of the unit square and describe fully the transformation a matrix represents.
I can find an unknown in a matrix or a point from a given image or area.

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