Transformations | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Describe and perform translations, reflections, rotationsHuraikan dan lakukan translasi, pantulan, putaran

Find and describe enlargements with scale factorCari dan huraikan pembesaran dengan faktor skala

Describe combined transformations as a single transformationHuraikan gabungan transformasi sebagai satu transformasi

Identify invariant points and lines of symmetryKenal pasti titik tak berubah dan garis simetri

📖 Key Concepts📖 Konsep Utama

Translation (Form 2)Translasi (Tingkatan 2)

Slides a shape without turning. Described by a column vector.Meluncurkan bentuk tanpa memutar. Dihuraikan oleh vektor lajur.

Translation vector: (p/q) → move p right, q up
Negative = left/down: (−3/2) = 3 left, 2 up

Reflection (Form 2)Pantulan (Tingkatan 2)

Mirror image across a line. Shape and size preserved.Imej cermin merentasi garis. Bentuk dan saiz dikekalkan.

Common mirror lines: x = k, y = k, y = x, y = −x
Distance from mirror = distance of image from mirror

Rotation (Form 2)Putaran (Tingkatan 2)

Turns a shape about a centre. Needs centre, angle and direction.Memutar bentuk tentang pusat. Perlu pusat, sudut dan arah.

Described by: Centre of rotation, Angle (e.g. 90°), Direction (clockwise/anticlockwise)
90° clockwise: (x,y)→(y,−x) | 90° anticlockwise: (x,y)→(−y,x)

Enlargement (Form 5)Pembesaran (Tingkatan 5)

Scales a shape from a centre by a scale factor k. Area scales by k².Menskalakan bentuk dari pusat dengan faktor skala k. Luas skala mengikut k².

✏️ Worked Examples✏️ Contoh Penyelesaian

Triangle PQR has vertices P(1,2), Q(3,2), R(3,5). Find the image after translation (4/−1).Segitiga PQR mempunyai bucu P(1,2), Q(3,2), R(3,5). Cari imej selepas translasi (4/−1).

P'(1+4, 2−1) = P'(5, 1)

Q'(3+4, 2−1) = Q'(7, 1)

R'(3+4, 5−1) = R'(7, 4)

P'(5,1), Q'(7,1), R'(7,4)

Describe the single transformation that maps triangle A(1,1), B(3,1), C(3,4) to A'(−1,1), B'(−3,1), C'(−3,4).Huraikan transformasi tunggal yang memetakan segitiga A(1,1), B(3,1), C(3,4) ke A'(−1,1), B'(−3,1), C'(−3,4).

x-coordinates change sign, y-coordinates unchanged

This is a reflection in the y-axis (x = 0)Ini ialah pantulan dalam paksi-y (x = 0)

Reflection in the y-axisPantulan dalam paksi-y

Shape with area 6 cm² is enlarged by scale factor 3. Find the area of the image.Bentuk berluas 6 cm² dibesarkan dengan faktor skala 3. Cari luas imej.

Area of image = k² × area of object

= 3² × 6 = 9 × 6 = 54 cm²

54 cm²

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Incomplete transformation descriptionsHuraian transformasi tidak lengkap

Reflection: must state the mirror line. Rotation: must state centre, angle AND direction. Translation: must give the column vector. Missing any = lost marks!Pantulan: mesti nyatakan garis cermin. Putaran: mesti nyatakan pusat, sudut DAN arah. Translasi: mesti berikan vektor lajur. Hilang mana-mana = markah hilang!

❌ Forgetting area scales by k², not kLupa luas skala mengikut k², bukan k

If scale factor = 2, area becomes 4× bigger (2²), NOT 2× bigger. Length scales by k, area by k², volume by k³.Jika faktor skala = 2, luas menjadi 4× lebih besar (2²), BUKAN 2×. Panjang skala mengikut k, luas mengikut k², isipadu mengikut k³.

❌ Wrong rotation directionArah putaran salah

Always specify clockwise or anticlockwise. 90° clockwise ≠ 90° anticlockwise!Sentiasa nyatakan ikut jam atau lawan jam. 90° ikut jam ≠ 90° lawan jam!

📝 SPM Practice📝 Amalan SPM

Paper 1 — MCQKertas 1 — Pilihan Berganda

Under translation (3/−2), the point (5, 4) maps toDi bawah translasi (3/−2), titik (5, 4) memetakan ke

(8, 2)

(8, 6)

(2, 2)

(2, 6)

The image of (3, 2) after reflection in x = −1 isImej (3, 2) selepas pantulan dalam x = −1 ialah

(−5, 2)

(5, 2)

(3, −2)

(−3, 2)

Shape enlarged by k = 2. If area = 8, image area =Bentuk dibesarkan k = 2. Jika luas = 8, luas imej =

A 90° anticlockwise rotation about origin maps (2, 3) toPutaran 90° lawan jam tentang asalan memetakan (2, 3) ke

(3, −2)

(−3, 2)

(−2, 3)

(2, −3)

Combined: reflection in y-axis then translation (2/1). Point (−3, 4) maps toGabungan: pantulan dalam paksi-y kemudian translasi (2/1). Titik (−3, 4) memetakan ke

(5, 5)

(−1, 5)

(1, 5)

(−5, 3)

Paper 2 — StructuredKertas 2 — Berstruktur

6 marks6 markah

Triangle T has vertices A(1,1), B(5,1), C(5,4).
(a) Find the image of T under enlargement with centre O(0,0) and scale factor 2. [3 marks]
(b) Find the area of the image if area of T = 6 cm². [3 marks]Segitiga T mempunyai bucu A(1,1), B(5,1), C(5,4).
(a) Cari imej T di bawah pembesaran berpusat O(0,0) dan faktor skala 2. [3 markah]
(b) Cari luas imej jika luas T = 6 cm². [3 markah]

Solution (a)Penyelesaian (a)

Multiply each coordinate by k=2

A'(2,2), B'(10,2), C'(10,8)

A'(2,2), B'(10,2), C'(10,8) [3 marks3 markah]

Solution (b)Penyelesaian (b)

Image area = k² × object area = 4 × 6

24 cm² [3 marks3 markah]

📋 Formula Summary📋 Ringkasan Formula

Translation:Translasi: Vector (p/q) → add to coordinates
Reflection:Pantulan: State mirror line (x=k, y=k, y=x, y=−x)
Rotation:Putaran: Centre + Angle + Direction
Enlargement:Pembesaran: Centre + Scale factor k
Length × k | Area × k² | Volume × k³