https://file.aiquickdraw.com/mathbot/file/17302242923242i18p2bb.jpg

The problem in the image requires us to identify the correct transformations applied to the figure in the coordinate plane. Let's analyze each of the options based on the graph.

### Analysis of Transformations
- **Figure A to Figure A'** (pre-image to image):
    - Observing the translation direction from each point, it appears that **Figure A** has been shifted **6 units down and 6 units left** to reach **Figure A'**.

#### Checking Each Option
1. **Option A**: \( T_{(-6, -6)}(x, y) = (x - 6, y - 6) \)
   - This matches a translation of 6 units left and 6 units down, which is correct.

2. **Option B**: "6 units down; 6 units left"
   - This is also accurate for the transformation observed.

3. **Option C**: "6 units up; 6 units right"
   - This does not match, as we see a shift in the opposite direction.

4. **Option D**: \( T_{(6, 6)}(x, y) = (x + 6, y + 6) \)
   - This would imply moving the figure 6 units up and 6 units right, which does not match the transformation.

5. **Option E**: \( T_{(-6, 6)}(x, y) = (x - 6, y + 6) \)
   - This option would suggest moving 6 units left and 6 units up, which does not match the transformation.

### Correct Answers
The correct answers are:
- **Option A**: \( T_{(-6, -6)}(x, y) = (x - 6, y - 6) \)
- **Option B**: "6 units down; 6 units left"

Would you like further details or have any questions on the steps?

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Here are 5 related questions for further understanding:
1. How do you determine the direction of a transformation based on the coordinates?
2. What are the steps to verify a translation in the coordinate plane?
3. How can you differentiate between translations, rotations, and reflections?
4. What would be the result if a point undergoes multiple translations sequentially?
5. Can translations affect the shape or size of a figure? Why or why not?

**Tip:** In transformations, always check both the x and y coordinates individually to confirm direction and distance.

Select all the rules that apply to the transformation shown in the image, where the pre-image is transformed to the image on the coordinate plane.

Analyze transformations on a coordinate plane by identifying the correct translation rules. This solution covers the steps for understanding how a pre-image is shifted to an image with specific transformations, such as translations by given units up, down, left, or right.

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