Describing a translation
A translation is an isometry that moves every point of an object the same distance in the same direction. There is no turning and no flipping — the object simply slides to a new position, so the image is congruent to the object.
A translation is described by a translation vector written as a column vector: (x) over y, usually shown as a column with the horizontal move x on top and the vertical move y below.
Key idea / 要点
Under the vector (x above y): move x units to the right (or left if x is negative) and y units up (or down if y is negative). A point (a, b) maps to (a + x, b + y).
Finding an image
To find the image of a point, add the vector components to the coordinates. This is quick and reliable arithmetic.
Worked example / 例题
Point P(2, 3) is translated by the vector with x = 4 and y = −1. Find the image P′.
New x-coordinate: 2 + 4 = 6.
New y-coordinate: 3 + (−1) = 2.
So P′ = (6, 2). The point moved 4 units right and 1 unit down.
To reverse a translation, use the opposite vector: change the sign of both x and y. So the reverse of (4 above −1) is (−4 above 1).
Remember / 记住
- A translation slides — no turning, no flipping.
- Vector top = horizontal (right +, left −).
- Vector bottom = vertical (up +, down −).
- Image of (a, b) is (a + x, b + y).