Transformations (IGCSE Maths)
Learn how to perform and describe all four transformations tested in IGCSE Maths: translations using column vectors, reflections in mirror lines, rotations about a centre, and enlargements with integer and fractional scale factors. Each type is explained with its required description, followed by worked examples on coordinate grids. At the bottom, the practice question generator builds from translations only (Starter) through to mixed "describe the single transformation" questions (Master), with auto-marked answers and instant feedback.
What Are Transformations?
A transformation changes the position, orientation, or size of a shape. The original shape is called the object and the transformed shape is called the image. Vertices on the object are labelled \(A, B, C, \ldots\) and the corresponding vertices on the image are labelled \(A', B', C', \ldots\)
There are four types of transformation in IGCSE Maths. Translations, reflections, and rotations all produce an image that is congruent to the object (same shape and size). An enlargement changes the size of the shape, so the image is similar to the object (same shape, different size). The angles of the shape never change under any transformation.
Core Ideas
How to Describe Each Transformation
Exam questions saying "Describe fully..." need the name plus every detail below. Missing any part loses marks.
| Type | You must state | What stays the same? |
|---|---|---|
| Translation | The column vector \(\left(\begin{array}{c} a \\ b \end{array}\right)\) | Shape, size, orientation |
| Reflection | The equation of the mirror line (e.g. \(y = x\), \(x = 3\)) | Shape, size (orientation flips) |
| Rotation | Angle of rotation, direction (clockwise or anti-clockwise), and centre of rotation | Shape, size |
| Enlargement | Scale factor (it can be fractional or negative) and the coordinates of the centre of enlargement | Shape, angles (size changes) |
1. Translations
Translation by \(\left(\begin{array}{c} 5 \\ 3 \end{array}\right)\)
The vector \(\left(\begin{array}{c} 5 \\ 3 \end{array}\right)\) means move 5 right and 3 up. Shape, size, and orientation are unchanged.
How to translate a shape
- Pick one vertex on the object.
- Move it by the vector: horizontal first, then vertical.
- Repeat for each remaining vertex.
- Join the new points to draw the image.
💡 Example 1: Perform
Translate \((1,2),(4,2),(4,5)\) by \(\left(\begin{array}{c} -3 \\ 4 \end{array}\right)\).
💡 Example 2: Describe
Vertex \((2,1) \to (-4,3)\). Describe the transformation.
\(x: -4-2=-6\), \(y: 3-1=2\)
Translation by \(\left(\begin{array}{c} -6 \\ 2 \end{array}\right)\).
2. Reflections
Reflection in the line \(x = 3\)
Each point is the same perpendicular distance from the mirror line, but on the opposite side.
How to reflect a shape
- Draw the mirror line on the grid.
- For each vertex, count the perpendicular distance to the mirror line.
- Plot the image vertex the same distance on the other side.
- Join the image vertices.
\(x\)-axis: \((x,y) \to (x,-y)\)
\(y\)-axis: \((x,y) \to (-x,y)\)
\(y=x\): \((x,y) \to (y,x)\)
\(y=-x\): \((x,y) \to (-y,-x)\)
💡 Example 3: Perform
Reflect \((1,3),(5,3),(3,6)\) in \(x = -1\).
Use \(x' = 2(-1) - x\):
\[\begin{aligned} (1,3) &\to (-3,3) \\ (5,3) &\to (-7,3) \\ (3,6) &\to (-5,6) \end{aligned}\]💡 Example 4: Describe
\((2,1)\), \((5,1)\), \((5,4)\) \(\to\) \((1,2)\), \((1,5)\), \((4,5)\).
Coordinates have swapped: \((x,y) \to (y,x)\).
Reflection in \(y = x\).
3. Rotations
Rotation 90° anti-clockwise about the origin
Each point follows \((x,y) \to (-y,x)\). The shape turns 90° anti-clockwise about the origin.
How to rotate a shape
- Mark the centre of rotation on the grid.
- For each vertex, draw a line from the centre to the vertex.
- Rotate that line by the given angle in the given direction.
- Mark the new position at the same distance from the centre.
- Join the image vertices. (Tracing paper makes this much easier.)
90° ACW: \((x,y) \to (-y,x)\)
180°: \((x,y) \to (-x,-y)\)
90° CW: \((x,y) \to (y,-x)\)
💡 Example 5: Perform
Rotate \((2,1),(5,1),(5,3)\) by 90° ACW about O.
💡 Example 6: Describe
\((1,0)\), \((3,0)\), \((3,2)\) \(\to\) \((0,-1)\), \((0,-3)\), \((2,-3)\).
\((1,0) \to (0,-1)\): pattern is \((x,y) \to (y,-x)\).
Rotation, 90° CW, centre (0,0).
4. Enlargements
Enlargement, scale factor 2, centre (0,0)
Rays from the centre pass through each vertex of the object and continue to the image. Every distance is doubled.
How to enlarge a shape
- Mark the centre of enlargement on the grid.
- Draw a ray from the centre through each vertex of the object.
- Multiply the distance along each ray by the scale factor to find the image vertex.
- Join the image vertices.
💡 Example 7: SF 2, centre O
Enlarge \((1,1),(3,1),(1,4)\) by SF 2, centre (0,0).
💡 Example 8: Fractional SF
Enlarge \((2,2),(8,2),(8,6),(2,6)\) by SF \(\dfrac{1}{2}\), centre O.
💡 Example 9: Describe (centre not at origin)
\((4,3)\), \((6,3)\), \((6,5)\) \(\to\) \((7,5)\), \((10,5)\), \((10,8)\).
Object base = 2, image base = 3, so the scale factor is \(\dfrac{3}{2}\).
Now use the vertex \((4,3) \to (7,5)\) to find the centre:
\[\begin{array}{rcl} C_x + \frac{3}{2}(4 - C_x) &=& 7 \\ 6 - \frac{1}{2}C_x &=& 7 \\ C_x &=& -2 \end{array}\] \[\begin{array}{rcl} C_y + \frac{3}{2}(3 - C_y) &=& 5 \\ 4.5 - \frac{1}{2}C_y &=& 5 \\ C_y &=& -1 \end{array}\]Enlargement, SF \(\dfrac{3}{2}\), centre \((-2,-1)\).
Negative scale factors
A scale factor can be negative. The rule does not change: you still measure from the centre to a vertex and multiply by the scale factor. What changes is the sign, so the measurement is taken in the opposite direction. The image therefore lands on the other side of the centre and comes out upside down (rotated by 180°), with its lengths multiplied by the size of the scale factor.
Enlargement, scale factor \(-2\), centre \((0,0)\)
Each ray runs from a vertex, through the centre, and out the far side. Lengths are doubled and the image is turned upside down.
💡 Example 10: Negative SF, perform
Enlarge \((2,1),(5,1),(5,3)\) by scale factor \(-1\), centre \((1,0)\).
From the centre, go the same distance the other way:
\[\begin{aligned} (2,1) &\to (0,-1) \\ (5,1) &\to (-3,-1) \\ (5,3) &\to (-3,-3) \end{aligned}\]Scale factor \(-1\) keeps the size, so this image is congruent to the object.
💡 Example 11: Negative SF, describe
\((2,2)\), \((4,2)\), \((4,3)\) \(\to\) \((-1,-1)\), \((-5,-1)\), \((-5,-3)\).
The image is upside down and twice as long, so the scale factor is \(-2\).
Use the vertex \((2,2) \to (-1,-1)\):
\[\begin{array}{rcl} C_x + (-2)(2 - C_x) &=& -1 \\ 3C_x - 4 &=& -1 \\ C_x &=& 1 \end{array}\] \[\begin{array}{rcl} C_y + (-2)(2 - C_y) &=& -1 \\ 3C_y - 4 &=& -1 \\ C_y &=& 1 \end{array}\]Enlargement, SF \(-2\), centre \((1,1)\).
One special case is worth knowing: an enlargement of scale factor \(-1\) about a point does exactly the same job as a rotation of 180° about that same point. Either description earns the marks, so if you spot a shape that has been turned upside down without changing size, you may call it whichever you find easier.
5. Combined Transformations
Apply two transformations in sequence, then describe the single equivalent transformation.
Reflect in \(x\)-axis, then reflect in \(y\)-axis
G (blue) reflects in the \(x\)-axis to give G' (yellow, dashed). Then G' reflects in the \(y\)-axis to give G'' (green). The single equivalent: rotation 180° about the origin.
💡 Example 12: Two reflections
\((1,1)\), \((4,1)\), \((4,2)\): reflect in the \(x\)-axis, then in the \(y\)-axis.
Step 1: \((x,y) \to (x,-y)\): \((1,-1)\), \((4,-1)\), \((4,-2)\)
Step 2: \((x,y) \to (-x,y)\): \((-1,-1)\), \((-4,-1)\), \((-4,-2)\)
Overall: \((x,y) \to (-x,-y)\). Rotation, 180°, centre (0,0).
🔑 Key Points
"Describe fully" = name + every required detail.
Translations, reflections, rotations: congruent. Enlargements: similar.
Reflect in \(y = x\): swap coordinates. \(y = -x\): swap and negate.
Learn the three origin rotation rules.
To find the centre of enlargement: draw rays through matching vertices.
A negative scale factor puts the image on the other side of the centre, upside down.
⚠️ Common Mistakes
Forgetting the direction for a rotation.
Writing words instead of a column vector.
Confusing \(y = 1\) with \(x = 1\) for mirror lines.
Calling a fractional enlargement "a reduction".
Dropping the minus sign from a negative scale factor, or measuring the rays the wrong way from the centre.
Not using tracing paper when available.