Chapter 12 II Transformations III ENHANCE (1)

CHAPTER 12 TRANSFORMATIONS III

CHAPTER 12 TRANSFORMATIONS III (Penjelmaan)12.1 Revision of the 4 types of transformations which you have learnt. Ulangkaji 4 jenis penjelmaan yang telah dipelajari

TRANSLATION (Translasi) All the points on a given plane move along a straight line by the same distance at the same direction.The shape, size and orientation remain the same.

It is written as A( A with a translation of .

Semua titik pada satah yang diberi bergerak dalam satu garis lurus dengan

jarak dan arah yang sama.Bentuk, saiz dan orientasi objek adalah sama.

Ia ditulis sebagai A( A dengan translasi . REFLECTION (Pantulan) All the points of an object are reflected in a line called the axis of reflection or line of reflection.

It is written as A(A with a reflection in the line ..

Semua titik pada suatu objek dipantulkan pada satu garis yang dipanggil paksi

pantulan atau garis pantulan.

Ianya ditulis sebagai A(A dengan pantulan pada garis .

ROTATION (Putaran)

All points on the object are rotated through a fixed angle at the same direction about a fixed point.

The direction of rotation is either clockwise or anticlockwise.

The fixed point about which the rotation takes place is called the centre of rotation.

A Rotation is determined by (a) the centre of rotation

(b) the angle of rotation

(c) the direction of rotation

Semua titik pada suatu objek diputarkan melalui satu sudut tetap pada arah yang sama dengan suatu titik tetap.

Arah pantulan itu sama ada secara arah jam atau lawan jam.

Titik tetap di mana berlakunya putaran ini dipanggil pusat putaran.

Putaran ditentukan melalui(a) pusat putaran

(b) sudut putaran

(c) arah putaran

ENLARGEMENT (Pembesaran)All points on the object move from a fixed point (the centre of enlargement) according to a fixed ratio. (the scale factor).Semua titik pada objek bergerak dari suatu titik tetap (pusat pembesaran) mengikut skala yang ditetaplan (faktor skala).

ExampleABCDE ( FGHIJ with a translation .ABCDE ( KLMNO with a reflection at the line x = 4.ABCDE ( PQRST with a rotation of 90o clockwise about the point (-1, 5).

ABCDE ( AUVWX with an enlargement at point A(1, 2) with a scale factor of 2.12.2 Combination of two types of transformations

Symbol of combination of 2 transformations

P(A) represents the image of point A under transformation P.

PQ represents transformation Q followed by transformation P.

QP represents transformation P followed by transformation Q.

P2 represents two consecutive transformations of P.

Skills assessed

To determine the image of a given point or shape under the combination of 2 transformations.

To find a single transformation which is equivalent to a combination of 2 given transformations.

Describe fully two consecutive transformations which map an object to its image. Calculate the area of the image (or object) under an enlargement.

12.2a To Find the Image of a Point under the Combination of Translation and Reflection1. T = Translation

P = Reflection at y = 1.

Find the image of A under (a) PT (b) TP.

2. T = Translation



3. T = Translation



4. T = Translation

P = Reflection at x = 2.


5. T = Translation

P = Reflection at x = -1.


6. T = Translation

P = Reflection at y = x.


12.2b To Find the Image of a Point under the Combination of Translation and Rotation

1. T = Translation

R = Rotation 90o clockwise about point PFind the image of A under (a) RT (b) TR.

2. T = Translation

R = Rotation 90o clockwise about point P.

Find the image of A under (a) RT (b) TR.

3. T = Translation

R = Rotation 90o anticlockwise about point P Find the image of A under (a) RT (b) TR.

4. T = Translation

R = Rotation 90o anticlockwise about point P Find the image of A under (a) RT (b) TR.

12.2c To Find the Image of a Point under the Combination of Reflection and Rotation

1. V = Reflection at y = 1. R = Rotation 90o clockwise about P.

Find the image of A under (a) RV (b) VR.

2. U = Reflection at y = 2.

R = Rotation 90o clockwise about P.

Find the image of A under (a) RU (b) UR.

3. P = Reflection at x = -1.

R = Rotation 90o clockwise about H.

Find the image of A under (a) RP (b) PR.

4. P = Reflection at x = 4.

R = Rotation 90o anticlockwise about O.

Find the image of A under (a) RP (b) PR.

12.3 Description of a transformation when the object and image are given Full description must be given by stating

(i) the type of transformation,

(ii) details of the transformation required.

Type of transformationsDetails requiredExample

Translation

Translation or

Translation

ReflectionAxis of reflectionA reflection in the line x = 2A reflection in the line y = 3

A reflection in the line y = x

RotationAngle of rotation (90o or 180o)Direction of rotation

(clockwise or anticlockwise)

Centre of rotationA rotation of 90o in the clockwise direction about the point (1,3).A rotation of 90o in the anticlockwise direction about the origin.

A rotation of 180o in the clockwise direction about the point K.

EnlargementScale Factor (, 2 or 3)

Centre of enlargementAn enlargement with scale factor 3 at the centre (4,1).An enlargement at the point (4,1) through a scale factor of 2.

In the following diagrams, II is the image of I under a transformation V. Describe in full, transformation V.ExampleExercise

1.

V = Translation

2

V =

3

V = Reflection at the line x = 2.

4

V =

5

V = 6

V =

7

V = Rotation of 90o clockwise at (0,0)

8

V =

9

V =

10

V =

11

V = Enlargement with scale factor 2

at (0, 0).12

V =

13

V =

14

V =

12.4 Description of the combined transformations involved when the object and image are given

In the following diagrams, II is the image of I under a transformation V followed by another transformation W. Describe in full, transformations V and W.

Note that there are many possible combinations.

ExampleExercise

1.

V = Translation

W = Enlargement with scale factor 2

at (-3, 1).

2

V =

W =

3


W = Enlargement with scale factor

at (3, 0).

4

V =

W =

5

V =

W = 6

V =

W =

7

V = Rotation of 90o clockwise at (2,0)

W = Enlargement with scale factor 2 at (1,-3)8

V =

W =

9

V =

W =

10

V =

W =

12.5 To calculate the area of the image (or object) under an enlargement with scale factor k

Use the formula

Or 12.5A

No.Area of ObjectScale Factor, kArea of Image

15 cm22225 = 20 cm2

212 cm23

336 cm2

417 unit21.5

52.5 unit24

6

264 cm2

74560 cm2

8

24 cm2

9272 cm2

10

3157.5 unit2

111.5108

12

42 cm22.5

13

52800 cm2

143152

1515 cm2

135 cm2

1624 cm2384 cm2

17108 cm227 cm2

12.5B In the diagrams below, A is the image of A under an enlargement with scale factor k.

3

Area of A = 48, k =

Area of A=

6

Area of A = 48, k =

Area of A =

7

Area of A = 927, k = 3

Area of A =

8

Area of A = 4.2, k = 3

Area of shaded region

= 4.232 4.2

= 33.69

Area of A = 21.6, k = 1.5


=

10

Area of A = 47 cm2, k =


=

11

Area of A = 21 cm2, k =


= 12

Area of A = 33.6 cm2, k =

Area of image A

=

12.6 Description of a single transformation which is equivalent to two combined transformations

Using the object and image given in the following diagrams, describe in full, a single transformation which is equivalent to the combination of the two given transformations.

Note that you are required to know the cases for combination of two isometric transformations of the same type only.

ExampleExercise

1.

V = Translation ,W = Translation

WV = Translation.

2

V = Translation , W = Translation

WV =

3


W = Reflection at the line x = -1.WV = Translation

4

V = Reflection at ______________ .

W = Reflection at ______________

WV =

5

V = Reflection at the line y = -2.

W = Reflection at the line y = 0.

WV = Translation

6

V = Reflection at the line y = 2.


WV =

7


W = Reflection at the line y = -1.

WV = Rotation 180o at the point (2, -1)8



WV =

9

V = Rotation 90o clockwise at (2,0)W = Rotation 180o clockwise at (-1,1)

WV = Rotation 90o anticlockwise at (-2,-2)10

V = Rotation 90o clockwise at (2,0)

W = Rotation 90o clockwise at (0,2)

WV =

12.7 Questions Based On Examination Format Question 1

(a) Transformation P represents a reflection at the line y = 2. Transformation T represents a translation. Transformation R represents a rotation of 90o in the anticlockwise direction about the point (5, 4).State the coordinates of the image of point (3, 1) under the following transformation:(i) P,(ii) TP,

(iii) RT.

(b) In Diagram 1, quadrilateral KLMN is the image of quadrilateral EFGH under a transformation V followed by another transformation W.

Describe in full

(i) transformation V , and

(ii) transformation W.(c) Given that quadrilateral KLMN represents an area of 88 unit2, find the area represented by quadrilateral EFGH.Model Answer For Question 1

(b)

(i) V = Reflection at the line y = -1.

[2 marks] - (get 1 mark if no axis of reflection or the reflection axis is wrong)

(ii) W = Enlargement with scale factor 2 at centre K.

[3 marks] - (get 1 mark if enlargement only,

get 2 marks if enlargement with scale factor 2 or enlargement at centre K.)

(c)

EMBED Equation.3

Question 2Object PointEDB

Image PointFAL

Table 1

(a) Table 1 shows three pairs of corresponding object and image points under the same translation. State the coordinates of

(i) point A,

(ii) point B.

(b) Point D is the image for point J under a reflection. State the image of point K under the same reflection.

(c) In Diagram 2, triangle JKL is the image of triangle DEF under a transformation V followed by another transformation WDescribe in full


(ii) transformation W.

(d) Given that triangle JKL represents an area of 169 unit2, find the area represented by triangle DEF.Question 3 (1998)(a) Transformation G represents a reflection at the line x = 1. Transformation H represents a translation. Transformation K represents a rotation of 90o in the anticlockwise direction about the point (3, 0).State the coordinates of the image of point (5, 2) under the following transformation:

(i) G,

(ii) HG,

(iii) KH.

[5 marks]

(b) In Diagram 3, triangle LMN is the image of triangle RMS under a transformation V and triangle LMN is also the object which maps to the image triangle LQP under a transformation W. Describe in full


(ii) transformation W.

(c) Given that triangle LMN has an area of 21 unit2, find the area of quadrilateral MQPN.[7 marks]Question 4 (1999)

The graph in Diagram 8 shows the quadrilaterals EFGH, JKLM and NPQR.(a) Transformation T represents a translation. Transformation V represents a reflection at the line y = 4. Transformation W represents a reflection at the y-axis.(i) State the coordinates of the image of point E under the translation T.(ii) State the coordinates of the image of point H under the reflection V.

(iii) Find the coordinates of the image of point J under the transformation WT.(iv) By recognizing the image of EFGH under the transformation WV, describe in full a single transformation which is equivalent to transformation WV.

[7 marks]

(b) NPQR is the image of JKLM under a transformation S.

(i) Describe in full transformation S .

(ii) If the area of JKLM is 17 unit2, calculate the area of NPQR.

[5 marks]

Question 5 (2000J)(a) Transformation T represents a translation and transformation P represents a reflection at the line y = -2.

State the coordinates of the image of point (4, 3) under the following transformation:

(i) T,

(ii) P,

(iii) TP.

[4 marks]

(b) In Diagram 10, triangle HJK is the image of triangle EFG under a transformation V and triangle LMN is the image of triangle HJK under a transformation W.

Describe in full

(i) transformation V ,

(ii) transformation W, and(iii) a single transformation which is equivalent to WV.

[8 marks]

Question 6 (2000)

The graph in Diagram 8 shows the quadrilaterals DEFG, DKJH and LMNH.

(a) Transformation S represents a translation. Transformation T represents a reflection at line DEK.


(i) S,

(ii) T,

(iii) ST.

[4 marks]

(b) Quadrilateral DKJH is the image of quadrilateral DEFG under a transformation V and quadrilateral LMNH is the image of quadrilateral DKJH under a transformation W.

Describe in full(i) transformation V,(ii) transformation W.[6 marks]

(c) If the area of quadrilateral DEFG is 12.7 square units, calculate the area of quadrilateral DKJH.

[2 marks]

Question 7 (2001J)(a) Transformation P represents a reflection at the line that passes through (0, 0) and (6, 6). Transformation T represents a translation.


(i) T,

(ii) PT,

(iii) TP.

[5 marks]

(b) In Diagram 8, triangle DEC is the image of triangle ABC under a transformation V and triangle DFG is the image of triangle DEC under a transformation W.

(i) Describe in full transformation V.(ii) Given that transformation W is an enlargement. State the centre and scale factor of the enlargement.(iii) Calculate the area of triangle ABC if the area of quadrilateral CEFG is 24 square units.

[7 marks]

Question 8 (2001)(a) Transformation M represents a translation. Transformation P represents a reflection at the line y = 2.

State the coordinates of the image of point (-3, 0) under the following transformation:

(i) P,

(ii) MM,

(iii) PM.

[5 marks]

(b) Diagram 9 shows quadrilateral ODEF, quadrilateral OFGH and quadrilateral OLKJ which are drawn on square grids.(i) Given that transformation Q is a reflection at line OFL and transformation R is a reflection at line OHJ. If quadrilateral ODEF experiences transformation RQ, describe in full a single transformation which is equivalent to RQ.

(ii) Given that quadrilateral OLKJ is the image of quadrilateral OFGH under a transformation W.(a) Describe in full transformation W.

(b) Calculate the area of the shaded region if the area of quadrilateral OFGH is 12.7 square units.[7 marks]

Question 9 (2002J)

The graph in Diagram 9 shows the triangles KLM, KRQ and PNM.(a) Triangle PNM is the image of triangle KLM under a transformation G whereas triangle KRQ is the image of triangle KLM under a transformation H.Describe in full

(i) transformation G,(ii) transformation H.(b) Given that transformation D is a reflection at the line y = 1 and transformation E is a reflection at the x-axis. (i) State the coordinates of

(a) the image of point M under the transformation D,

(b) the image of point R under the transformation DE. (ii) If transformation W is a single transformation which is equivalent to transformation DE, describe in full transformation W.(c) Given that the area of triangle KLM is 6.3 square unit, calculate the area of the image of triangle KLM under an enlargement with a scale factor of 5. [7 marks]

Question 10(2002)

(a) Transformation R represents a rotation of 90o in the anti-clockwise direction at point (1, 4). Transformation P represents a reflection at the line y = 2.


(i) R,

(ii) PR.

[3 marks]

(b) The graph in diagram 8 shows quadrilaterals A, B, C and D.

(i) Quadrilateral B is the image of quadrilateral A under a transformation V, whereas quadrilateral C is the image of quadrilateral B under a transformation W.

Describe in full

(a) transformation V,(b) a single transformation which is equivalent to transformation WV.

(ii) Quadrilateral D is the image of quadrilateral A under a certain enlargement. (a) State the scale factor of the enlargement.

(b) Find the coordinates of the centre of the enlargement.

(c) If the area of quadrilateral A is 7.5 square units, calculate the area of quadrilateral D.[9 marks]

ANSWERS

Chapter 12 Transformations III12.2a

2 (a) A(7, 2), A(7, 0) (b) A(1, -1), A(7, -2)3 (a) A(-2, 5), A(-2, 1) (b) A(2, 4), A(-2, 7)4 (a) A(3, 1), A(1, 1) (b) A(0, -1), A(-1, 1)5 (a) A(1, 2), A(-3, 2) (b) A(2, 4), A(5, 2)6 (a) A(3, 1), A(1, 3) (b) A(3, -2), A(8, -4)

12.2b

2 (a) A(6, 3), A(4, -1) (b) A(3, 0), A(7, 2)3 (a) A(1, 2), A(3, 0) (b) A(6, 2), A(4, 5)4 (a) A(5, 1), A(3, 1) (b) A(5, 2), A(4, 4)

12.2c

1 (b) A(5, 4), A(5, -2)2 (a) A(3, 1), A(-2, 0) (b) A(0, 0), A(0, 4)3 (a) A(-3, 2), A(1, 4) (b) A(1, 0), A(-3, 0)4 (a) A(5, 1), A(-1, 5) (b) A(-1, 3), A(9, 3)

12.3

2 Translation

3 Reflection at x = 24 Reflection at x = 0

5 Reflection at y = 1

6 Reflection at y = x

7 Rotation of 90o clockwise about (0, 0).

8 Rotation of 90o clockwise about (0, -1).

9 Rotation of 180o clockwise about (1, 1).

10 Rotation of 180o clockwise about (1, 0.5).

11 Enlargement with scale factor 2 at (0, 0)

12 Enlargement with scale factor 2 at (1, 4)

13 Enlargement with scale factor 3 at (-3, 0)

14 Enlargement with scale factor at (-1, 1).

12.4

2 V = Translation

W = Enlargement with scale factor 3 at (0, 3)4 V = Reflection at the line x = 1

W = Enlargement with scale factor 2 at (-1, 0)5 V = Reflection at the line y = - 2

W= Enlargement with scale factor 2.5 at (3, -2)6 V = Reflection at the line y = - 1

W = Enlargement with scale factor 2 at (4, -1)8 V = Rotation of 90o anti-clockwise about (0, -2).

W = Enlargement with scale factor 2 at (4, 0)

9 V = Rotation of 90o clockwise about (3, 1).

W = Enlargement with scale factor 2 at (2, -1)

10 V = Rotation of 180o clockwise about (2, -2).

W = Enlargement with scale factor 2.5 at (3, -2)

12.5A

2. 108

3. 9

4. 38.25

5. 40

7. 35

8. 96

9. 18

10. 17.5

11. 48

12. 262.5

13. 112

14. 1260

16. 4

17. 0.5

12.5B

2. 31.5

3. 12

5. 68

6. 192

7. 103

9. 27

10. k=2, 141

11. k=3, 168

12. k=2, 134.4

12.6

2. V = Translation

W = Translation

WV = Translation

4. V = Reflection at x = 0

W = Reflection at x = 4 WV = Translation

6. WV = Translation

8. WV = Rotation 180o clockwise about (3,1)

10. WV = Rotation 180o clockwise about (0,0)

12.7 Questions According to Examination Format

2(a) (i) A(1, 3) (ii) B(2, 0)

(b) (3, -4)

(c) V = Rotation 90o anti-clockwise about point F.

W = Enlargement with scale factor 2 at point J.

(d) 42.25

3(a) (i) (-3, 2) (ii) (-1, -4) (iii) (7, 4)

(b) (i) V = Rotation 180o clockwise about point M.

(ii) W = Enlargement with scale factor 3 at point L.

(a) 168

4(a) (i) (-1, 8) (ii) (2, 3) (iii) (5, 4)

(iv) WV = Rotation 180o clockwise about point (0, 4).

(b) (i) Enlargement with scale factor 3 at point (-2, 1).

(ii) 153

5(a) (i) (-2, 6) (ii) (4, -7) (iii) (-2, - 4)

(b) (i) V = Rotation 90o anti-clockwise about point (4, 0).

(ii) W = Rotation 90o clockwise about point (1, -1).

(iii) Translation

6(a) (i) (-1, 5) (ii) (3, 4) (iii) (1, 7)

(b) (i) V = Enlargement with scale factor 2 at point D.

(ii) W = Rotation 90o anti-clockwise about point H.

(c) 50.8

7(a) (i) (5, 1) (ii) (1, 5) (iii) (4, 2)

(b) (i) V = Rotation 90o clockwise about point C.

(ii) Centre D, scale factor = 2

(iii) 8

8 (a) (i) (-3, 4) (ii) (5, 2) (iii) (1, 3)

(b) (i) RQ = Rotation 90o clockwise about point O.

(ii) (a) W = Enlargement with scale factor 3 at point O.

(b) 101.6

9. (a) (i) Rotation 180o clockwise about point M.

(ii) Enlargement with scale factor 3 at point K.

(b) (i) (a) (6, 2) (b) (-4, -1)

(ii) W = Translation.

(c) 157.5

10. (a) (i) (4, 6) (ii) (4, -2)

(b) (i) (a) V = Reflection at the line x = 5

(b) WV = Rotation 180o clockwise about point (5, 4)

(ii) (a) 3 (b) (6, 7) (c) 67.5

(a) A(1,3)A( , ) A( , )

(b) A(1,3)A( , ) A( , )

(a) A(2, 2)A( , ) A( , )

(b) A(2, 2)A( , ) A( , )

(a) A(2,4)A(5,3) A(5,-1)

(b) A(2,4)A(2,-2) A(5,-3)

(a) A(4,-1)A( , ) A( , )

(b) A(4,-1)A( , ) A( , )

(a) A(-2,4)A( , ) A( , )

(b) A(-2,4)A( , ) A( , )

(a) A(-2,3)A( , ) A( , )

(b) A(-2,3)A( , ) A( , )

A(2,3)A(5,2) A(2,-3)

(b) A(2,3)A(3,0) A(6,-1)

DIAGRAM 2

(a) A(2,1)A( , ) A( , )

(b) A(2,1)A( , ) A( , )

(a) A(3,-1)A( , ) A( , )

(b) A(3,-1)A( , ) A( , )

(a) A(6,-1)A( , ) A( , )

(b) A(6,-1)A( , ) A( , )

A(2,3)A(2,-1) A(1,4)

(b) A(2,3)A( , ) A( , )

3

(a) A(3,3)A( , ) A( , )

(b) A(3,3)A( , ) A( , )

(a) A(3,1)A( , ) A( , )

(b) A(3,1)A( , ) A( , )

(a) A(1,2)A( , ) A( , )

(b) A(1,2)A( , ) A( , )

-6

x

0

-2

-4

4

2

3

2

-1

O

y

x

6

0

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-2

-3

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

DIAGRAM 8

I

II

I

II

O

I

II

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

II

-3

-2

II

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

1

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

II

I

2

O

4

O

A

B

A

B

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

4

-1

2

1

3

1

x

y

3

2

O

I

II

II

I

II

I

II

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

-3

-2

-1

-4

-3

-2

5

3

1

x

y

3

2

O

I

II

I

II

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

II

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

I

P

II

-1

-2

-3

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

II

II

I

I

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

II

I

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

O

II

B

A

I

A

B

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

II

I

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

II

I

2

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

II

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

-8

I

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

3

2

O

II

I

II

-4

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

-2

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

I

6

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

O

III

y

4

I

2

II

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

-5

4

I

O

2

3

y

x

1

3

1

2

-1

4

5

-2

-3

-4

-1

-2

-3

y

I

x

O

I

-3

-2

-1

-4

-3

-2

5

4

-1

2

1

3

1

x

y

2

O

3

III

II

III

I

II

III

III

I

III

8

-6

6

0

-2

-4

4

2

-4

-2

III

II

-6

K

L

M

N

E

F

G

H

8

Q

-6

10

0

-2

1

-4

-3

-3

-1

-1

-5

1

3

5

D

E

F

J

L

K

DIAGRAM 1

4

P

8

x

2

-2

N

N

S

DIAGRAM 3

M

DIAGRAM 10

R

N

L

2

2

4

6

y

Q

R

N

L

K

J

M

E

F

G

H

6

L

Q

0

-3

4

2

1

3

5

7

x

y

3

-1

-1

-2

-3

-4

-5

4

5

J

K

H

F

G

E

M

N

L

K

A

E

H

J

N

D

A

B

L

M

-8

y

x

6

4

-2

-4

-6

2

0

8

6

2

4

DIAGRAM 8

Q

F

G

D

K

10

R

A

Q

B

DIAGRAM 9

G

A

F

DIAGRAM 8

E

D

C

Q

M

N

P

8

F

G

E

D

L

K

O

J

H

Area of Image = k2 Area of Object

y

6

4

2

Q

-4

-6

-2

0

-2

-4

4

2

N

x

DIAGRAM 9

Area of Object = EMBED Equation.3

Q

C

4

2

y

x

16

14

8

6

12

0

8

6

2

4

DIAGRAM 8

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

-4

A

A

A

-2

-4

4

2

4

-2

2

(i) (3, 3) 1 mark

(ii) (0, 5) 2 marks

(iii) (6, -1) 2 marks

y =2

G

H

F

E

N

M

L

K

-6

-4

-2

6

4

2

y

x

8

-6

6

0

-2

-4

4

2

PAGE 2Transformations III

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Documents

Chapter 12 II Transformations III ENHANCE (1)