Let's select a point:
Point A (2, 3)
Now, let's perform two transformations:
- Reflection over the line y = x
- Translation by (4, -2)
Performing the transformations in the first order
Step 1: Reflect Point A (2, 3) over the line y = x.
The rule for reflecting a point (x, y) over the line y = x is to switch the coordinates:
\( (x, y) \rightarrow (y, x) \)
So:
\[
\text{Reflection of } A: (2, 3) \rightarrow (3, 2)
\]
Step 2: Now translate the reflected point (3, 2) by (4, -2).
The rule for translation is given by adding the translation vector to the coordinates.
\[
(x, y) \rightarrow (x + a, y + b)
\]
where \( (a, b) \) is the translation vector.
For our translation vector (4, -2):
\[
(3, 2) \rightarrow (3 + 4, 2 - 2) \rightarrow (7, 0)
\]
Final point after the first order of transformations:
Point A' (7, 0)
Performing the transformations in the second order
Now, let's switch the order and do the translation first, followed by the reflection.
Step 1: Translate Point A (2, 3) by (4, -2).
\[
(2, 3) \rightarrow (2 + 4, 3 - 2) \rightarrow (6, 1)
\]
Step 2: Reflect the translated point (6, 1) over the line y = x.
Using the reflection rule again:
\[
(6, 1) \rightarrow (1, 6)
\]
Final point after the second order of transformations:
Point A'' (1, 6)
Summary of Results
- After performing reflection first and then translation, the final point is (7, 0).
- After performing translation first and then reflection, the final point is (1, 6).
These results show that the order in which transformations are applied significantly affects the final position of the point.