A reflection needs the word and the mirror line. A rotation needs the word, the angle with its direction, and the centre. Leaving any of these out gives an incomplete description.
This lesson follows describing a translation with a column vector and builds on recognising the effect of reflecting a graph.
What exactly is needed for each transformation?
| Transformation | Must state |
|---|---|
| Translation | The word and the column vector |
| Reflection | The word and the equation of the mirror line |
| Rotation | The word, the angle, clockwise or anticlockwise, and the centre |
A reflection flips a shape across a line, and each point is as far behind the line as its object is in front. A rotation turns the shape about a fixed point, and each point stays the same distance from that centre.
How do I find the details from a diagram?
- Match one point on the object to its image.
- For a reflection, find the midpoint of the two points. The mirror line goes through it, at right angles to the line joining them.
- For a rotation, check the turn: compare the direction from the centre to the object with the direction to the image.
- Write the full description with every required detail.
Worked example
Triangle ABC has A(1, 2), B(4, 2) and C(4, 5). Its image is A′(5, 2), B′(2, 2) and C′(2, 5). Describe the transformation completely.
Step 1, which type? The image is the same size but flipped left to right: A is on the left of B, but A′ is on the right of B′. That points to a reflection.
Step 2, midpoints: A and A′ give ((1 + 5) ÷ 2, 2) = (3, 2). B and B′ give ((4 + 2) ÷ 2, 2) = (3, 2). C and C′ give (3, (5 + 5) ÷ 2) = (3, 5).
Step 3, the line: all midpoints have x = 3, so the mirror line is vertical.
Answer: reflection in the line x = 3.
Check: distance of A from the line is 3 − 1 = 2, and A′ is 5 − 3 = 2 on the other side. Correct.
The mistake to watch for
Two slips happen a lot with rotations. One is leaving out the centre. The other is turning the wrong way.
Mistaken answer: P(3, 1) is rotated 90° anticlockwise about the origin. The student wrote the image as (1, −3).
This uses the rule for a clockwise turn.
A 90° anticlockwise turn about the origin maps (x, y) to (−y, x). So P(3, 1) becomes P′(−1, 3).
A quick sanity check: P is in the first quadrant, and an anticlockwise quarter turn moves it into the second quadrant, where x is negative and y is positive. The mistaken (1, −3) is in the fourth quadrant.
Check yourself
Try these, then open each answer.
1. Reflect the point P(2, 5) in the line y = 1. Find its image.
Show answer
P is 5 − 1 = 4 above the line, so the image is 4 below it: 1 − 4 = −3. The x value stays 2.
P′ = (2, −3)
2. Rotate the point (3, −2) through 180° about the origin.
Show answer
A half turn changes both signs: (x, y) becomes (−x, −y).
(−3, 2)
3. Rotate (2, 0) through 90° clockwise about the origin.
Show answer
Clockwise 90° maps (x, y) to (y, −x). So (2, 0) becomes (0, −2).
(0, −2)
Check: it is a quarter turn clockwise from the positive x axis, which lands on the negative y axis.
Where this leads next
Try the vectors and transformations practice set, where transformations are mixed with vectors. For the notation behind positions, read position versus displacement. The non-calculator working trainer supports the signed arithmetic.
When a description is nearly right but one detail is always missing, our teachers can build a checking routine with you in online one-to-one Mathematics tuition.