Transformations.
Track corresponding vertices and give the complete instruction for each move.
Reflect, rotate, enlarge and translate · use fractional and negative factors · combine transformations in order
Match each vertex to its image
Imagine cutting a triangle from card and moving it on a grid. Sliding, turning and flipping change where its vertices appear, while stretching changes its size. A transformation is a precise instruction for one of these changes.
Name the starting shape the object and the result the image. Match each original vertex to its corresponding image vertex before comparing the shapes. The explorer below keeps those matches visible, so you can connect a movement with the instruction needed to reproduce it.
Use corresponding vertices to identify what changed. Reflections, rotations and translations preserve lengths and angles. Enlargements preserve angles and multiply lengths by the absolute scale factor.
Reflection in x = 0 (y-axis).
| Vertex | Original | Image |
|---|---|---|
| A | (1, 1) | (-1, 1) |
| B | (3, 1) | (-3, 1) |
| C | (1, 2) | (-1, 2) |
Pause and explainWhy is saying “move the triangle” an incomplete transformation description?
It does not specify which movement or how far, around which centre, or across which mirror line it should occur.
Reflect and rotate about the correct line or centre
A reflection is a flip across a mirror line. A rotation is a turn around a fixed centre. They can place a shape in similar-looking positions, but they relate corresponding points in different ways.
For reflection, compare perpendicular distances on opposite sides of the mirror. For rotation, compare turns and distances from the centre. Origin coordinate rules are shortcuts for specific standard cases; understanding the geometry tells you when those shortcuts are valid.
For reflection, each point and its image are equal perpendicular distances from the mirror line, on opposite sides. Points on the mirror line stay fixed.
| Transformation | (x, y) maps to |
|---|---|
| Reflection in x-axis | (x, −y) |
| Reflection in y-axis | (−x, y) |
| Reflection in y = x | (y, x) |
| Reflection in y = −x | (−y, −x) |
| 90° anticlockwise about origin | (−y, x) |
| 90° clockwise about origin | (y, −x) |
| 180° about origin | (−x, −y) |
Rotate P (3, 2) by 90° anticlockwise about C (1, 1).
Work relative to the centre: P − C = (2, 1). Rotate that displacement to (−1, 2), then add C back.
Pause and explainWhich point stays fixed in every rotation about a given centre?
The centre itself. Its distance from the centre is zero, so turning does not move it.
Enlargement can reduce or reverse a shape
An enlargement moves every vertex along a line through a centre, scaling its directed distance from that centre. The centre, rather than the origin automatically, is the reference point for the movement.
A factor larger than 1 increases lengths; a positive factor below 1 reduces them. A negative factor places the image on the opposite side of the centre as well. Follow one point at a time: find its displacement from the centre, scale it, then return to coordinates.
Draw a line from the centre through each vertex. Multiply the directed displacement from the centre by k. A positive k keeps the image on the same side; a negative k puts it on the opposite side.
Enlarge P (3, 2) from centre C (1, −1) with scale factor −2.
Displacement P − C = (2, 3). Multiply by −2 to get (−4, −6).
For k = 12, all corresponding lengths halve. For k = −12, they halve and move to the other side of the centre. A negative enlargement is not a reflection in a line.
Pause and explainDoes “enlargement” always mean a larger image?
No. A factor between 0 and 1 makes a smaller image, while negative factors also reverse the side of the centre.
Translate and combine in the stated order
A translation slides every point by exactly the same horizontal and vertical changes. It does not turn, flip or resize the shape. The column vector records those two changes in order.
For combinations, treat the first image as the object for the next instruction. Record intermediate coordinates rather than trying to guess one final movement. Reversing the order can change the result because the second operation acts on different starting positions.
Translate (−2, 5) by 3−4.
Top component changes x; bottom component changes y. Translation has no centre.
Reflect P (2, 1) in the x-axis, then translate by 32.
Reflection gives (2, −1), then translation gives (5, 1).
Reversing the order gives (5, 3) then (5, −3). The two combinations are different.
Pause and explainWhy does a translation vector have the same components for every vertex?
A slide moves all points by one common displacement, preserving the shape and its orientation.
Your turn. Give a complete description.
Use corresponding vertices and give enough information for someone else to draw exactly the same image. Name mirror lines, centres, angles, directions, factors and vectors as needed.
For a combination, show the intermediate image. Check one other vertex afterward to confirm that your instruction describes the whole shape rather than only one conveniently matched point.
Reflect P (4, −2) in the line x = 1.
Reveal worked answer
Rotate (−2, 3) by 90° clockwise about the origin.
Reveal worked answer
Enlarge (4, −2) from the origin with scale factor −12.
Reveal worked answer
Give the translation vector that maps A (−1, 3) to A′ (4, −2).
Reveal worked answer
A shape maps to another after a turn of 180° about (2, −1). Describe the transformation fully.