Question

two students apply different sequences to the same point (2, -1):
student a: rotate 90° clockwise, then translate 3 units left
student b: translate 3 units left, then rotate 90° clockwise
calculate the final position for both students. show your work. explain why the order of transformations matters.
question 6
i can identify a sequence of rigid transformations by analyzing a figure and its image.
i can do this by myself and can explain it to someone else.

Explanation:

Step1: Calculate Student A's final position

• Rotation:
• The rule for a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\).
• For the point \((2,-1)\), after rotation, it becomes \((-1,-2)\).
• Translation:
• The rule for translating 3 units left is \((x,y)\to(x - 3,y)\).
• For the point \((-1,-2)\), after translation, it becomes \((-1-3,-2)=(-4,-2)\).

Step2: Calculate Student B's final position

• Translation:
• The rule for translating 3 units left is \((x,y)\to(x - 3,y)\).
• For the point \((2,-1)\), after translation, it becomes \((2 - 3,-1)=(-1,-1)\).
• Rotation:
• The rule for a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\).
• For the point \((-1,-1)\), after rotation, it becomes \((-1,1)\).

Step3: Explain why the order of transformations matters

• Rotation and translation are non - commutative operations.
• A rotation changes the direction of the vector from the origin, and a translation shifts the point. When we rotate first and then translate, the translation is in the coordinate system after rotation. When we translate first and then rotate, the rotation is about the origin of the original coordinate system, and the translation is in the original coordinate system.

Answer:

Student A's final position is \((-4,-2)\). Student B's final position is \((-1,1)\). The order of transformations matters because rotation and translation are non - commutative operations.