Translation shifts the origin from O(0,0) to a new origin O'(h,k) without changing the orientation of the axes. The coordinate transformation is X=x-h, Y=y-k (equivalently, x=X+h, y=Y+k).
Primary purpose: removing first-degree terms from the equation of a conic. For the general conic ax^{2+2hxy+by}^{2+2gx+2fy+c}=0 (without xy-term, i.e., h=0), translation to the center (-g/a, -f/b) eliminates the linear terms 2gx and 2fy. With the xy-term present, the center is found by solving the system: ax+hy+g=0, hx+by+f=0 simultaneously.
Key properties preserved by translation: distances between points, angles between lines, areas of regions, the shape and size of all curves. Translation only changes the position of the reference point.
After translation, the conic equation takes a simpler standard form. For example, x^{2+y}^{2-4x+6y-12}=0 translates to X^{2+Y}^2=25 by shifting to center (2,-3). The circle's equation is now in standard form, immediately revealing center (at the new origin) and radius 5.
For parabolas, which have no center, translation is to the vertex. The equation =0 becomes (y+2)^2=8(x-1), and translating to (1,-2) gives =8X, the standard form.
Translation is also used as the second step (after rotation) in reducing the general second-degree equation to standard form. Once the xy-term is removed by rotation, the resulting equation has linear terms that can be eliminated by translating to the appropriate center or vertex.