Reflections, Rotations and Translations
Transformations describe how a shape moves while preserving its size and shape. The key GCSE skill is not only constructing an image, but also describing a transformation completely from an object–image pair.
Foundation & Higher, with a Higher extension. You should be comfortable with coordinates in four quadrants and with line notation such as x = 2, y = −1 and y = x. Linear Graphs is useful background.
What transformations preserve
Reflections, rotations and translations are isometries: corresponding lengths and angles stay the same, so area is unchanged.
A reflection reverses orientation. A rotation and a translation preserve orientation.
| Transformation | Must be stated |
|---|---|
| Reflection | Mirror line |
| Rotation | Centre, angle and direction |
| Translation | A two-component vector |
Reflections
To reflect a point, the mirror line must be the perpendicular bisector of the segment joining the original point to its image.
For a reflection in x = 2, a point at x = 5 lies 3 units to the right of the mirror, so its image lies 3 units to the left at x = −1. The y-coordinate stays unchanged.
For reflections in y = x, the coordinates swap: (a,b) → (b,a). For y = −x, swap and change both signs: (a,b) → (−b,−a).
Rotations
A rotation needs three details: centre, angle and direction. The centre is fixed, and every point stays the same distance from it.
For example, a 90° clockwise rotation about (2,1) is not the same as rotating 90° clockwise about the origin. Mark the centre first, then rotate each vertex around it.
Worked example — rotate one vertex
Rotate A(5,2) 90° clockwise about C(2,1). Relative to C, A is 3 right and 1 up. A clockwise quarter-turn changes that to 1 right and 3 down. Add that movement to C: A′=(2+1,1−3)=(3,−2). Repeat for every vertex. A half-turn reverses both relative movements.
To recover a rotation centre, join corresponding vertices and construct perpendicular bisectors of two joining segments. Their intersection is the centre.
Translations and column vectors
A translation moves every point by the same vector. The top number is the horizontal movement; the bottom number is the vertical movement.
For the column vector , move 4 units left and 3 units up. Negative horizontal values move left; negative vertical values move down. In text, the same components may be written (−4, 3), horizontal first.
Translation-vector notation begins here. Vector arithmetic is developed later in Vectors and Vector Arithmetic.
Recovering the transformation
Compare corresponding points rather than guessing from the overall picture.
| Clue | Likely transformation |
|---|---|
| All points move by the same vector | Translation |
| Each point is equally far from a mirror line on the opposite side | Reflection |
| All points stay the same distance from one fixed centre | Rotation |
A full description must contain every required parameter. ‘A rotation’ is incomplete; ‘rotation 90° clockwise about (1,−2)’ is complete.
Higher — combining transformations
When two transformations are performed, order matters. Reflect then translate can produce a different final image from translate then reflect.
Track an intermediate image rather than trying to do both mentally. If the final mapping is recognisable as one familiar transformation, you may describe it, but not every composition collapses neatly to a single reflection, rotation or translation.
A fixed point is a point that does not move under a transformation. Rotation centres are fixed points; points on a mirror line are fixed by that reflection.
Think Like a Mathematician
Describe transformations from invariant structure: equal distances to a mirror, equal distances from a centre, or one common displacement vector. Those properties are more reliable than visual appearance.
Common Mistakes
Reflection: moving every point by the same amount instead of using equal perpendicular distance from the mirror.
Rotation: assuming the centre is the origin.
Translation: writing vertical movement first in the vector.
Combinations: assuming the order never matters.
Practice — transformations
1. Reflect (5,−2) in the y-axis.
The x-coordinate changes sign: (−5,−2).
2. Reflect (−3,4) in the x-axis.
The y-coordinate changes sign: (−3,−4).
3. Reflect (2,−5) in the line y=x.
Swap coordinates: (−5,2).
4. Translate (−1,3) by vector (4,−6).
(−1+4, 3−6) = (3,−3).
5. What information is needed to describe a rotation fully?
Centre, angle and direction.
6. A point is 3 units right of the line x=1. Where is its reflected image relative to the line?
3 units left of the line.
7. A triangle is reflected in y=−x. What happens to point (4,1)?
(x,y)→(−y,−x), so (−1,−4).
8. A translation sends (2,5) to (−3,8). Find the vector.
Horizontal: −5. Vertical: +3. Vector (−5,3).
9. A 180° rotation about (0,0) sends (4,−7) where?
(−4,7).
10. Why is ‘rotation 90°’ incomplete?
It does not state the centre or direction.
11. Higher: if a shape is reflected in x=0 then translated by (2,0), is that always the same as translating first then reflecting?
No. For example, start at (1,0). Reflect in x=0 then translate by (2,0): (1,0)→(−1,0)→(1,0). Reverse the order: (1,0)→(3,0)→(−3,0). The endpoints differ.
12. State one fixed-point fact for a reflection.
Every point lying on the mirror line is fixed.
Frequently Asked Questions
Does a reflection change area?
No. Reflections preserve lengths, angles and area.
Can a rotation centre be outside the shape?
Yes. The centre can be anywhere in the plane.
Are combined transformations Foundation?
The core single transformations are Foundation and Higher. Combinations are taught here as a Higher extension; both tiers need complete descriptions of the single transformations.
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