. Final answer
The final mapping of vertices of ADEF is A'(2, -2), D'(4, 2), E'(5, 1), and F'(9, 5).
Explanation
The mapping of the vertices ADEF involves applying the equation y = x to each point to find their corresponding positions. By using the given coordinates of D, E, and F, and applying the equation y = x, the coordinates of the new vertices A', D', E', and F' are determined.
Point D(2, 4) becomes D'(4, 2) as the x and y coordinates swap according to y = x. Similarly, E(1, -1) translates to E'(1, -1) and F(5, 1) becomes F'(5, 1). A, however, is not provided but can be inferred as it mirrors D across the line y = x, resulting in A'(2, -2). Therefore, the complete mapping of ADEF is A'(2, -2), D'(4, 2), E'(1, -1), and F'(5, 1).
The transformation follows the rule of reflecting points across the line y = x, effectively swapping their x and y coordinates. This geometric transformation is based on the given equation y = x, which dictates that for any point (x, y), the new position after the transformation will be (y, x). Each vertex's coordinates are recalculated according to this rule, allowing us to determine their new positions in the plane.
This method ensures that the mapping maintains consistency and precision in translating the vertices from their initial positions to their final mapped locations.