Slide to the New Coordinate

A point can land in exactly the right new place without turning, flipping, or changing shape. The secret is hidden in two tiny instructions. Read them in the right order, and you can predict the landing spot before the point even moves.

Coordinates always read across first, then up or down. At point A, the first coordinate controls the horizontal move from zero. The second controls the vertical move. Trace an L-shaped path: across to three, then up to two, so A is three, two. Never swap that order. Translation uses a vector: the top number changes the horizontal coordinate, and the bottom number changes the vertical coordinate. Positive means right or up. Negative means left or down. Add each instruction to its matching coordinate. Keep x with x, and y with y. Watch the point slide along the grid. A translation does not mirror it, spin it, or swap its coordinates. Every part travels the same distance in the same direction. For one point, simply mark the new landing position.

Let's work through an example. On a coordinate grid, point P is at two, three. It is translated by the vector four, negative two. What are the coordinates of the image, P prime? Start with horizontal. The x-coordinate is two, and the vector's horizontal instruction is positive four. Add two and four to get six. On screen, P slides four squares to the right. The new x-coordinate is six. Now take the vertical instruction. The y-coordinate is three, and the vector says negative two. Add three and negative two to get one. So P moves two squares down. It does not move up, because the sign is negative. Put the two new coordinates back in their usual order: horizontal first, vertical second. P prime is six, one. Check the move from P: four right, two down. That matches the vector exactly, so six, one is the answer.

Across first, vertical second, then add the vector in matching order. Slide the point, check the direction... and you have translated the coordinate with confidence!