Understanding Transformation of Functions Using Matrices: Rotations, Translations, Dilations, and Reflections 🌍

Welcome to this exciting lesson on **transformations of functions using matrices**! 🎉 In Maths, a transformation is a way to change the position, size, or orientation of a shape or a point on a coordinate plane. Think of it like moving, resizing, or flipping an object. We'll explore four main types of transformations: **dilations** [scaling], **reflections** [flipping], **rotations** [turning], and **translations** [sliding]. What's really cool is that we can represent these changes using special tools called **matrices**! Using matrices makes complex transformations much simpler and allows us to perform multiple transformations efficiently.

What Are Transformations? 🤔

In geometry, a **transformation** is a function that maps points of a shape to new points, resulting in a new shape [called the image]. Every point in the original shape [called the pre-image] corresponds to a unique point in the image. We'll focus on transformations in a two-dimensional [2D] coordinate system.

• **Translation**: Moving a shape from one location to another without changing its size, shape, or orientation. It's simply a "slide." ↔️
• **Dilation [Scaling]**: Changing the size of a shape by stretching or shrinking it. This transformation can make an object bigger or smaller. ↕️
• **Reflection**: Flipping a shape over a line [called the line of reflection] to create a mirror image. 🪞
• **Rotation**: Turning a shape around a fixed point [called the center of rotation] by a certain angle. 🔄

Representing Points and Transformations with Matrices 📝

In a 2D coordinate system, a point $(x, y)$ can be represented as a **column matrix** [also known as a column vector]:

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

A **transformation matrix** is a square matrix that, when multiplied by a point's column matrix, produces the coordinates of the transformed point. For 2D transformations around the origin, we typically use a $2 \times 2$ matrix.

If $P = \begin{bmatrix} x \\ y \end{bmatrix}$ is the original point and $T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is the transformation matrix, then the new point $P' = \begin{bmatrix} x' \\ y' \end{bmatrix}$ is found by matrix multiplication:

$$P' = T \cdot P \Rightarrow \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

This multiplication gives us the new coordinates:

$$x' = ax + by$$ $$y' = cx + dy$$

Specific Transformation Matrices 🛠️

1. Dilation [Scaling]

A dilation changes the size of a figure. If we want to scale a point by a factor of $k_x$ in the $x$-direction and $k_y$ in the $y$-direction, the transformation matrix is:

$$D = \begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix}$$

If $k_x = k_y = k$, it's a **uniform dilation** [scaling by a single factor $k$]. If $k > 1$, the figure gets larger; if $0 < k < 1$, it gets smaller.

The new point will be $(k_x x, k_y y)$.

2. Reflection 🪞

Reflections flip a figure over a line. Here are common reflection matrices:

• **Reflection across the x-axis:** $$R_x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$ The point $(x, y)$ becomes $(x, -y)$.
• **Reflection across the y-axis:** $$R_y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$$ The point $(x, y)$ becomes $(-x, y)$.
• **Reflection across the line $y=x$:** $$R_{y=x} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ The point $(x, y)$ becomes $(y, x)$.
• **Reflection across the line $y=-x$:** $$R_{y=-x} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$$ The point $(x, y)$ becomes $(-y, -x)$.

3. Rotation 🔄

Rotations turn a figure around a fixed point [usually the origin]. The standard rotation matrix for a counter-clockwise rotation by an angle $\theta$ around the origin is:

$$R_\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$

Common rotation matrices:

• **Rotation by $90^\circ$ counter-clockwise [CCW]** $(\theta = 90^\circ$ or $\frac{\pi}{2}$ radians): $$(R_{90^\circ} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$) The point $(x, y)$ becomes $(-y, x)$.
• **Rotation by $180^\circ$ counter-clockwise [CCW]** $(\theta = 180^\circ$ or $\pi$ radians): $$R_{180^\circ} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$ The point $(x, y)$ becomes $(-x, -y)$.
• **Rotation by $270^\circ$ counter-clockwise [CCW]** $(\theta = 270^\circ$ or $\frac{3\pi}{2}$ radians), which is the same as $90^\circ$ clockwise [CW]: $$R_{270^\circ} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$ The point $(x, y)$ becomes $(y, -x)$.

4. Translation ↔️

A translation slides a figure without changing its orientation or size. Unlike the other transformations, a translation cannot be achieved solely by a $2 \times 2$ matrix multiplication in the standard coordinate system. Instead, we perform a vector addition to the point's coordinates after any matrix transformation.

If a point $(x, y)$ is translated by a vector $\begin{bmatrix} h \\ k \end{bmatrix}$, the new point $(x', y')$ is:

$$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} h \\ k \end{bmatrix} = \begin{bmatrix} x+h \\ y+k \end{bmatrix}$$

So, a translation vector $\begin{bmatrix} h \\ k \end{bmatrix}$ shifts points $h$ units horizontally and $k$ units vertically. A positive $h$ means right, negative $h$ means left. A positive $k$ means up, negative $k$ means down.

Combining Transformations 🧩

When you perform multiple transformations, the **order matters**! Each transformation is applied sequentially. If you apply transformation $T_1$ first, then $T_2$, and then $T_3$ to a point $P$, the final transformed point $P'$ is calculated as:

$$P' = T_3 \cdot (T_2 \cdot (T_1 \cdot P))$$

Due to the associative property of matrix multiplication, this can also be written as:

$$P' = (T_3 \cdot T_2 \cdot T_1) \cdot P$$

This means you can multiply the transformation matrices together [in reverse order of application] to get a single combined transformation matrix. However, remember that translations are additions and must be handled separately after all matrix multiplications.

Detailed Examples 💡

Time for a Quick Check! 🧠

Test your understanding with these interactive questions.

Lesson Summary 🏆

You've successfully journeyed through the world of **function transformations using matrices**! We covered the four main types of transformations and how to represent and apply them mathematically:

• **Dilation**: Changes the size of a shape and is represented by a scaling matrix like $D = \begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix}$.
• **Reflection**: Flips a shape across a line, with specific matrices for reflections across axes or lines like $y=x$. For example, $R_x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ for reflection across the x-axis.
• **Rotation**: Turns a shape around a fixed point [typically the origin] by an angle $\theta$, using a general rotation matrix $R_\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$. Key rotations like $90^\circ$ CCW have simplified matrices like $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.
• **Translation**: Slides a shape without changing its orientation or size. This is handled by **vector addition** $\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} h \\ k \end{bmatrix}$, rather than a $2 \times 2$ matrix multiplication.

Remember that the **order of transformations matters**! When combining multiple matrix transformations, you multiply the matrices in the reverse order of application. However, translations are always added after all matrix multiplications. Mastering these concepts provides a powerful tool for understanding and manipulating geometric shapes! Keep practicing, and you'll soon be a transformation pro! ✨