Advanced Transformations
Beyond simple slides and flips, advanced transformations explore what happens when shapes shrink or flip through a central point. This guide covers enlargements with fractional and negative scale factors, and combinations of transformations — key skills for the higher-tier GCSE Maths exam.
The Core Concepts
Enlargements with Tricky Scale Factors
Fractional scale factor (e.g., $\frac{1}{2}$): The enlargement makes the shape smaller. The image appears on the same side of the centre of enlargement as the original.
Negative scale factor (e.g., −2): The enlargement makes the shape bigger and flips it to the opposite side of the centre of enlargement. The image is also rotated 180°.
Combinations of Transformations
When you apply more than one transformation to a shape, the key rule is:
Perform the transformations one at a time. Apply the first to the original, then apply the second to the resulting image.
Worked Examples
Example 1: Fractional Enlargement
Enlarge triangle A by scale factor $\frac{1}{2}$, centre $(1,0)$.
Each vertex of the image is half the distance from the centre compared to the original.
Example 2: Negative Enlargement
Enlarge square ABCD by scale factor −2, centre $(0,0)$.
The image will be twice as large and appear on the opposite side of the origin, rotated 180°.
Tutor Insights
🤔 Common Misunderstandings
- Negative scale factors: Forgetting that the image appears on the opposite side of the centre, rotated 180°.
- Finding the centre: When describing an enlargement, draw lines through corresponding vertex pairs — they all meet at the centre.
- Order of transformations: Assuming the order doesn’t matter. Reflect then translate is usually different from translate then reflect.
📝 Common Exam Mistakes
- Using the origin as the centre for enlargements when a different centre is stated.
- Incorrectly applying the scale factor to vectors from the centre of enlargement.
- Not providing a full description of a transformation — always include the centre for enlargements and rotations.
Practice Questions
- Enlarge triangle A with vertices at $(2,4)$, $(6,4)$, and $(2,8)$ by a scale factor of $\frac{1}{2}$ about the origin $(0,0)$.
- Rectangle C has vertices at $(1,1)$, $(4,1)$, $(4,3)$, and $(1,3)$. Enlarge it by a scale factor of $-1$ about the centre $(2,2)$.
- Triangle E has vertices at $(1,2)$, $(3,2)$, and $(2,4)$. First, reflect E in the line $y = x$ to get image F. Then translate F by $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$ to get image G. What are the coordinates of G?
Show Answers
- Answer: $(1, 2)$, $(3, 2)$, $(1, 4)$.
- Answer: $(3, 3)$, $(0, 3)$, $(0, 1)$, $(3, 1)$.
- Working: Reflecting in $y = x$ swaps coordinates → F: $(2,1)$, $(2,3)$, $(4,2)$. Translating by $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$:
Answer: G has vertices $(-1, 2)$, $(-1, 4)$, $(1, 3)$.
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