In geometry, Translation means to move each point the same distance, and in the same direction. This fixed direction with a fixed distance is known as a "vector." Under translation, all points move along a straight line parallel to the translation vector.
In the interactive worksheet above the translation vector is shown as a green arrow. It is the directed segment OV, starting at the origin (0,0) of the XY-plane, and ending at point V.
As you move around point V, each red point is translated, from the position of the corresponding blue point, in the direction indicated by the translation vector OV (the green arrow). The distance of this motion equals the length of the translation vector, that is, the distance between the origin (0,0) and point V.
Translations are examples of geometrical transformations called isometries, because they preserve not only the shapes of geometric figures but also their size.
Translations in general are not linear transformations because translations move the origin (0,0) to a different point, V, the ending point of the translation vector. Linear transformations, on the contrary, do not move the origin (0,0), they keep it in the same place.
The formula defining a translation is that of vector addition. For example, if point V has coordinates (h,k), then:
Q = F(P) = F(x,y) = (x,y) + (h,k) = (x+h,y+k)
where P represents any of the blue points A1,..,D1; and Q represents the corresponding red, translated point A2,..,D2.
In the interactive GeoGebra worksheet applet presented above, you can move the blue points around the screen, not only one at a time but also two at a time by segments, meaning, you can move each one of A1B1, B1C1, C1D1, and D1A1, the sides of the blue polygon.
I want to thank Kenneth Frank for preparing a Turkish version of this Translation transformation GeoGebra applet.