Combination of Transformations

 

What Are Combined Transformations?

A combined transformation is when two or more transformations are performed on a shape, one after the other.

The final image may be the result of multiple steps, but sometimes these can be simplified to a single transformation.

 

Common Transformations

Here’s a quick reminder of the four main transformations:

• Translation – shifts a shape using a vector.
• Reflection – flips a shape across a mirror line.
• Rotation – turns a shape around a point.
• Enlargement – scales a shape from a centre point.

Each of these can be combined to form more complex movements.

 

Important Notes:

• The order of transformations matters.
• Two transformations can sometimes be simplified into a single equivalent transformation.
• Some combinations produce well-known results (like two reflections making a rotation).

 

Example

Question: A triangle is rotated $180\degree$ about the origin. The image is then reflected in the x-axis.

 

 

Describe the single transformation that maps the original triangle to the final image.

Answer: Rotation $180\degree, \quad \to$  flips the shape to the opposite quadrant. Then reflection in the x-axis $\to$ flips it again.

The combined effect is a reflection in the y-axis.

Final Answer: Reflection in the y-axis.

 

 

Example

Question: On a grid, triangle A is translated by the vector: $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$

The image is then reflected in the line $y = 1$.

Describe the single transformation that maps triangle A to the final image.

 

 

Solution

Step 1: Translate

• Move each vertex 4 right and 3 down.
• This gives the intermediate shape (we can call it A').

Step 2: Reflect in $y = 1$

• Use the line $y = 1$ as the mirror line.
• Reflect A' over this line to get A'.

Now describe the whole process as a single transformation.

 Final Answer: 

No single transformation can exactly match both a translation and a reflection. So, the correct answer is:

A translation followed by a reflection in the line $y = 1$

 

 

 

 

Key Reversal Rules

 

From A to B	Reverse from B to A
Translation by $\begin{pmatrix} x \\ y \end{pmatrix}$	$\begin{pmatrix} -x \\ -y \end{pmatrix}$
Reflection in a line	Reflection in the same line
Rotation $\theta\degree$ clockwise	Rotation $\theta\degree$ anticlockwise
Enlargement SF = $k$	Enlargement SF = $\frac{1}{k}$