IB Mathematics AI – Topic 3

General Form:

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} e \\ f \end{pmatrix} \]

Components:

• \(\begin{pmatrix} x \\ y \end{pmatrix}\): Original point coordinates
• \(\begin{pmatrix} x' \\ y' \end{pmatrix}\): Image point coordinates (after transformation)
• \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\): Transformation matrix (handles rotations, reflections, stretches)
• \(\begin{pmatrix} e \\ f \end{pmatrix}\): Translation vector (handles shifts)

Process for Multiple Points:

1. Write coordinates as column vectors
2. Multiply by transformation matrix
3. Add translation vector
4. Result gives image coordinates

Notation:

Often written as: \(\mathbf{x'} = A\mathbf{x} + \mathbf{b}\)

Where A is the transformation matrix and b is translation vector

⚠️ Common Pitfalls & Tips:

• Order matters: matrix multiplication is NOT commutative
• Always apply matrix transformation BEFORE translation
• Use GDC for calculations - faster and more accurate
• Can transform multiple points at once using matrix with multiple columns

Common Reflection Matrices:

1. Reflection in x-axis:

\[ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]

Effect: (x, y) → (x, -y)

2. Reflection in y-axis:

\[ \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \]

Effect: (x, y) → (-x, y)

3. Reflection in line y = x:

\[ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

Effect: (x, y) → (y, x) - swaps coordinates

4. Reflection in line y = -x:

\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \]

5. Reflection in line y = x tan θ:

\[ \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} \]

Formula in IB formula booklet

⚠️ Common Pitfalls & Tips:

• Reflection matrices have determinant ±1
• Reflecting twice returns to original position
• For diagonal reflections, use formula booklet
• Check which axis/line is the mirror

Rotation Matrix (Anticlockwise about origin by angle θ):

\[ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]

In IB formula booklet

Common Rotation Angles:

90° anticlockwise:

\[ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

180°:

\[ \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]

270° anticlockwise (or 90° clockwise):

\[ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]

Clockwise Rotation:

For clockwise rotation by angle θ, use -θ in the formula:

\[ \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \]

⚠️ Common Pitfalls & Tips:

• Default rotation is anticlockwise from positive x-axis
• For clockwise, use negative angle
• Rotation is always about the origin unless stated otherwise
• Check calculator mode (degrees or radians)

1. Stretch parallel to x-axis (scale factor k):

\[ \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \]

Effect: (x, y) → (kx, y)

2. Stretch parallel to y-axis (scale factor k):

\[ \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \]

Effect: (x, y) → (x, ky)

3. Enlargement (scale factor k from origin):

\[ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \]

Effect: (x, y) → (kx, ky)

This scales uniformly in all directions

Properties:

• If k > 1: enlargement (stretches away from axis/origin)
• If 0 < k < 1: reduction (shrinks toward axis/origin)
• If k < 0: stretch/enlargement with reflection

⚠️ Common Pitfalls & Tips:

• Diagonal matrices represent stretches/enlargements
• Determinant = product of diagonal elements = k₁ × k₂
• Area scale factor = |determinant|
• Negative scale factor includes reflection

Translation Vector:

\[ \begin{pmatrix} e \\ f \end{pmatrix} \]

Added to the result of matrix transformation

Effect:

Shifts point by e units horizontally and f units vertically

\[ (x, y) \to (x + e, y + f) \]

Combined with Matrix Transformation:

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = A\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} e \\ f \end{pmatrix} \]

The translation is applied AFTER the matrix transformation

⚠️ Common Pitfalls & Tips:

• Translation is represented by a vector, not a 2×2 matrix
• Always add translation AFTER matrix multiplication
• Translation does not change shape or size
• Cannot be represented by 2×2 matrix alone

Combining Transformations:

To apply transformation B followed by transformation A:

\[ \text{Combined matrix} = AB \]

Important: Apply from right to left (B first, then A)

Area Scale Factor:

When a shape is transformed by matrix A, the area is scaled by:

\[ \text{Area of image} = |\det(A)| \times \text{Area of original} \]

Key Facts About Determinants:

• |det(A)| = 1: Area preserved (reflections, rotations)
• |det(A)| > 1: Area increases (enlargements)
• |det(A)| < 1: Area decreases (reductions)
• det(A) = 0: Shape collapses to line or point

Finding Original Point:

If you know the image and transformation, find original using:

\[ \mathbf{x} = A^{-1}(\mathbf{x'} - \mathbf{b}) \]

⚠️ Common Pitfalls & Tips:

• Order matters: AB ≠ BA in general
• Use absolute value of determinant for area
• Translations don't affect area (not in the matrix)
• Use GDC to find determinants and inverses

Solution:

Step 1: Write point as column vector

\[ \mathbf{x} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

Step 2: Apply matrix transformation

\[ A\mathbf{x} = \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix}\begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

\[ = \begin{pmatrix} 2(3) + (-1)(2) \\ 1(3) + 3(2) \end{pmatrix} = \begin{pmatrix} 6 - 2 \\ 3 + 6 \end{pmatrix} = \begin{pmatrix} 4 \\ 9 \end{pmatrix} \]

Step 3: Add translation vector

\[ \mathbf{x'} = \begin{pmatrix} 4 \\ 9 \end{pmatrix} + \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \end{pmatrix} \]

Answer: P'(8, 8)

Solution:

Step 1: Calculate determinant of transformation matrix

\[ \det(A) = (3)(4) - (1)(2) = 12 - 2 = 10 \]

Step 2: Apply area formula

Area scale factor = |det(A)| = |10| = 10

\[ \text{Area of image} = |\det(A)| \times \text{Original area} \]

\[ = 10 \times 12 = 120 \text{ cm}^2 \]

Answer: 120 cm²

General Form

• \(\mathbf{x'} = A\mathbf{x} + \mathbf{b}\)
• Matrix first, then translate
• Use GDC for calculations

Area Scale Factor

• Factor = |det(A)|
• Reflections/Rotations: |det| = 1
• New area = factor × old area

Key Matrices

• All in formula booklet
• Check angles (deg/rad)
• Order matters for combinations

Finding Original

• \(\mathbf{x} = A^{-1}(\mathbf{x'} - \mathbf{b})\)
• Subtract translation first
• Then multiply by inverse