How does the order matter when making rules for translations, rotations, reflections and dilations? Select a point (neither the x-value or y-value may be 0). Write down two different types of rules (translations, rotations, reflections and dilations) and perform them in one order. What is your final point? Use the same original point and the same two rules but switch the order. What is your final point? Show all of your steps.
When making rules for translations, rotations, reflections, and dilations, the order matters because each transformation is applied based on the original coordinates of the point. Changing the order in which the transformations are applied will yield different final points.
Let's use the point P(2, 3) for our examples.
Example 1:
Translation: Move the point 4 units to the right and 2 units up.
Rotation: Rotate the point 90 degrees counterclockwise about the origin.
Steps:
1. Translation: P(2, 3) + (4, 2) = (6, 5)
2. Rotation: Rotate (6, 5) 90 degrees counterclockwise about the origin:
New x-coordinate = x*cos(90°) - y*sin(90°) = 5
New y-coordinate = x*sin(90°) + y*cos(90°) = -6
Therefore, the final point is (-5, 6).
Example 2:
Rotation: Rotate the point 90 degrees counterclockwise about the origin.
Translation: Move the point 4 units to the right and 2 units up.
Steps:
1. Rotation: Rotate P(2, 3) 90 degrees counterclockwise about the origin:
New x-coordinate = x*cos(90°) - y*sin(90°) = -3
New y-coordinate = x*sin(90°) + y*cos(90°) = 2
2. Translation: (-3, 2) + (4, 2) = (1, 4)
Therefore, the final point is (1, 4).
As shown, switching the order of the transformations resulted in different final points.