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.
The order does matter when making rules for translations, rotations, reflections, and dilations because the operations are not commutative. Changes in order can result in different outcomes.
Let's select the point (3, 4) as our original point.
First, let's perform a translation by adding the vector (2, 2) to the original point:
(3, 4) + (2, 2) = (5, 6)
Next, let's perform a dilation with a scale factor of 2 centered at the origin:
(5, 6) * 2 = (10, 12)
So, the final point after performing the translation and dilation in that order is (10, 12).
Now, let's switch the order of the rules.
First, let's perform the dilation with a scale factor of 2 centered at the origin:
(3, 4) * 2 = (6, 8)
Next, let's perform the translation by adding the vector (2, 2) to the previous point:
(6, 8) + (2, 2) = (8, 10)
So, the final point after performing the dilation and translation in that order is (8, 10).
In conclusion, by switching the order of the rules from the first example, we obtained a different final point. This illustrates that the order of operations does matter when making rules for translations, rotations, reflections, and dilations.