Question

A certain mapping in the xy-plane has the following properties: Every point moves m units horizontally and n units vertically. If m is positive, the point moves to the right. If m is negative, the point moves to the left. If n is positive, the point moves up. If n is negative, the point moves down. Which transformation does the mapping define?

Option 1: A translation
Option 2: A reflection
Option 3: A rotation

Answer 1

The mapping described represents a translation, which moves every point of a shape the same distance in the same direction without changing its orientation. If m is positive, the translation is to the right; if m is negative, it's to the left. Similarly, if n is positive, the translation is upward, and if n is negative, downward.

Step-by-step explanation:

The mapping described in the question specifies a transformation where every point in the xy-plane moves horizontally (left or right) and vertically (up or down), based on the values of m and n respectively. This type of transformation in geometry is known as a translation. A translation moves every point of a shape or figure the same distance in the same direction. In this case, if m is positive, the translation is to the right; if m is negative, it's to the left. Similarly, if n is positive, the translation is upward, and if n is negative, it's downward.

Translations do not change the orientation of the shape being moved; they simply slide the entire shape m units horizontally and n units vertically, depending on the signs of m and n. This is distinct from a reflection, which flips a shape over a line or a rotation, which pivots a shape around a point. Thus, the transformation defined by the mapping is Option 1: A translation.