It looks like you've uploaded an image containing a series of geometric transformation exercises, including rotations, reflections, translations, and some vector-related problems. Let me guide you through solving a few of these problems.

1. Question 6 - Use your rules to solve these transformations

For example:

• (a) $T_{2,-1}(A(5,4))$ involves a translation by 2 units right and 1 unit down. To compute: $A'(5 + 2, 4 - 1) = A'(7, 3)$

• (d) $T_{-3, 5}(A(4,-3))$: $A'(4 - 3, -3 + 5) = A'(1, 2)$

2. Rotation problem example

• $R_{90^\circ}(A(3, 2))$: A 90-degree rotation about the origin involves swapping the coordinates and changing the sign of the new x-coordinate. $A'(x', y') = (-y, x) = (-2, 3)$

3. Reflection Problem

• Reflect over the y-axis, $R_y(A(5,-1))$: Reflections over the y-axis reverse the sign of the x-coordinate. $A'(5, -1) \rightarrow A'(-5, -1)$

Would you like help with any specific problem from this set?

Follow-up Questions:

1. How do you determine coordinates after a 180-degree rotation?
2. Can you describe how translation affects the x and y coordinates of a point?
3. How would reflecting over the x-axis change the coordinates of a point?
4. What is the geometric effect of a 270-degree rotation?
5. How can you find the smallest positive angle for a given rotation matrix?

Tip: In transformations, always pay attention to the order of operations when combining multiple steps (e.g., reflect and then translate).