The value a controls horizontal displacement, while b controls vertical displacement. A positive value moves points in the corresponding positive coordinate direction, and a negative value moves them in the opposite direction. A zero value leaves that coordinate unchanged, so the pair (a, b) specifies the complete displacement applied to every point.
Every point receives the same displacement vector, so the coordinate difference between any two points stays the same. Because those relative differences do not change, distances and angles remain unchanged. Parallel lines remain parallel as well, and the figure keeps its orientation, shape, and size after the mapping.
The rule (x, y) → (x + a, y + b) lets a numerical operation predict a visible movement. Adding a to each x-coordinate and b to each y-coordinate produces the coordinates of the image. Plotting those results reveals how the algebraic rule changes the figure in space.
First, identify the translation vector and list the coordinates of each vertex. Next, add the vector components to every vertex using the same rule. Record the resulting image coordinates, then plot or connect them in the original order. Consistent application ensures that the entire polygon receives the intended displacement.
Compare an original point with its corresponding image point and calculate the horizontal and vertical coordinate changes. If the same pair of changes appears for every corresponding point, those values form the translation vector. This check helps distinguish a consistent translation from a mapping that changes points by different amounts.
Translations support coordinate-geometry problems by converting a spatial movement into an algebraic rule. They can model motion, simplify spatial relationships, and help analyze figures without changing their essential geometric properties. Comparing original and image coordinates also connects visual reasoning with calculations, making the method useful for studying mapped figures.