The map Figure M onto Figure N, we need to perform the following transformations:
Reflection across the x-axis. This flips Figure M upside down.
Rotation 180 degrees clockwise about the origin. This turns Figure M upside down and around.
Translation 3 units upward. This moves Figure N up so that it is centered on the y-axis.
The steps above can be summarized as follows:
R_180 * R_x * T_up(3)
where:
R_180 is a rotation of 180 degrees clockwise about the origin
R_x is a reflection across the x-axis
T_up(3) is a translation 3 units upward
Here is a diagram that illustrates the transformations:
M -> R_x(M) -> R_180(R_x(M)) -> T_up(3)(R_180(R_x(M))) = N
We can also describe the transformations using coordinate geometry. Let (x,y) be a point in Figure M.
Then, the following transformations will map (x,y) to the corresponding point (x ′ ,y') in Figure N:
x' = x
y' = -y - 3
This can be written as a single matrix transformation as follows:
[[1, 0], [0, -1]] * [x; y] + [0; -3]
This matrix transformation first reflects the point (x,y) across the x-axis, then rotates it 180 degrees clockwise about the origin, and finally translates it 3 units upward.