A translation slides every point of a shape the same distance in the same direction, without turning or flipping it. To describe one completely, write the word translation and its column vector.
This skill opens vectors and transformations and connects to translating a simple graph.
What does a column vector tell me?
A column vector has two numbers, one above the other. The upper number is the move across (x direction) and the lower number is the move up or down (y direction).
Right and up are positive. Left and down are negative.
In text on this site we write a column vector as (3, −1), meaning 3 in the upper row and −1 in the lower row.
Points are written with a capital letter, such as P(3, −1), so you can tell a point from a vector.
How do I find the vector from an object and its image?
- Pick one point on the object and the corresponding point on the image.
- Subtract image minus object for the x values, then for the y values.
- Write the two answers as a column, x above.
- Check by adding the vector to the object point. You should land on the image point.
Worked example
Point P(−2, 3) is mapped to P′(4, −1) by a translation. Describe the translation.
Step 1, x movement: image x minus object x = 4 − (−2) = 4 + 2 = 6.
Step 2, y movement: image y minus object y = −1 − 3 = −4.
Step 3, write the answer: the translation has column vector (6, −4).
Step 4, check: −2 + 6 = 4 and 3 + (−4) = −1. The object lands on P′, so the vector is right.
In words, that is 6 units right and 4 units down. The vector is the full description, and the same vector moves every other point of the shape too.
The mistake to watch for
A frequent slip is subtracting the wrong way round, object minus image.
Mistaken answer: (−6, 4)
The student worked −2 − 4 = −6 and 3 − (−1) = 4, which is the movement from P′ back to P.
That vector is a genuine translation, but it is the reverse one.
The fix is to think “where I end minus where I start”. A quick check is to add your vector to the object point. Here −2 + (−6) = −8, not 4, so the check fails straight away.
Check yourself
Try these, then open each answer.
1. Point Q(−5, 2) is translated by the vector (4, −7). What are the coordinates of the image?
Show answer
x: −5 + 4 = −1. y: 2 + (−7) = −5.
Q′ = (−1, −5)
2. A translation maps A(1, −3) to A′(−2, 5). Write its column vector.
Show answer
x: −2 − 1 = −3. y: 5 − (−3) = 8.
Vector (−3, 8). Check: 1 + (−3) = −2 and −3 + 8 = 5.
3. A translation has vector (2, 5). What vector takes the image back to the object?
Show answer
Reverse both movements: (−2, −5).
Where this leads next
Next, learn to add displacement vectors on a diagram, since two translations in a row give one combined vector. When you are ready, use the vectors and transformations practice set. The non-calculator working trainer helps with the signed arithmetic.
Some students can draw a translation perfectly but lose marks on the vector notation. That is the kind of detail our teachers watch for in online one-to-one Mathematics tuition.