Final answer:
A translation of XY to XY' in a coordinate plane is expressed as the rule (x, y) → (x + Δx, y + Δy), where Δx and Δy are the specified distances in the horizontal and vertical directions, respectively.
Step-by-step explanation:
When we say that XY' is a translation of XY, we're referring to a type of transformation in the coordinate system where every point of the object is moved the same distance in the same direction. In mathematics, especially in vector arithmetic, this rule can be written as a function applied to each coordinate.
To express this translation as a rule for a point (x, y), we would denote the translation as:
(x, y) → (x + Δx, y + Δy)
where Δx represents the change in the x-coordinate (how far the point moves horizontally) and Δy represents the change in the y-coordinate (how far the point moves vertically). If Δx is positive, the translation is to the right; if Δx is negative, to the left. Similarly, if Δy is positive, the translation is upward, and if Δy is negative, it is downward.
The specific values of Δx and Δy would depend on the particular translation being applied to the object XY to produce XY'.