Question

(a)
Use the information for Part A and Part B


Part A
Which statement describes a possible sequence of transformations that transforms figure 1 into figure 2?

A
a reflection across the x-axis, followed by a translation 2 units to the left
B
a rotation 180 degree clockwise about the origin, Followed by a translation 3 units to the right
C
a reflection across the x-axis, followed by a translation 3 units to the right
D
a rotation 180 degree clockwise about the origin, Followed by a translation 2 units to the left
(b)

Part B
Figure 3 can also be created by transforming figure 1 with a sequence of two transformations.
​
A
a rotation 90 degree clockwise about the origin, Followed by translation 3 units to the right
B
a rotation 90 degree clockwise about the origin, Followed by reflection across the x-axis
C
a rotation 180 degree clockwise about the origin, Followed by reflection across the y-axis
D
a rotation 180 degree clockwise about the origin, Followed by a translation 2 units to the left

​

Answer 1

To determine the possible sequences of transformations for both Parts A and B, let’s analyze each option based on common transformations in geometry.

### Part A:

To go from figure 1 to figure 2, we evaluate each transformation option:

**A**: A reflection across the x-axis, followed by a translation 2 units to the left.
- **Reflection across the x-axis** will invert the y-coordinates, which might change the position significantly depending on the figure. Then moving 2 units left would depend on the initial new positions of the points resulting from the reflection.

**B**: A rotation 180 degrees clockwise about the origin, followed by a translation 3 units to the right.
- **Rotation 180 degrees** would effectively invert the coordinates (changes both the x and y signs). After this transformation, translating right changes the x-coordinate positively. This could fit certain figures quite well.

**C**: A reflection across the x-axis, followed by a translation 3 units to the right.
- The first transformation will invert the y-coordinates, then moving 3 units to the right could work depending on the position of the resulting points.

**D**: A rotation 180 degrees clockwise about the origin, followed by a translation 2 units to the left.
- Similar to option B, a rotation would change the signs of the x and y coordinates, and moving 2 units left would be a linear transformation in the opposite direction in the x-axis.

**Conclusion for Part A:**
Without seeing the actual figures, it's difficult to definitively select the appropriate sequence, but generally, options B and D are typically the most accurate choices when it comes to significant transformations like rotations followed by translations.

### Part B:

For transforming figure 1 into figure 3, we analyze the options:

**A**: A rotation 90 degrees clockwise about the origin, followed by a translation 3 units to the right.
- A 90-degree rotation would change the coordinates in a way (x, y) → (y, -x). Afterward, translating right would keep the figure in the new space.

**B**: A rotation 90 degrees clockwise about the origin, followed by reflection across the x-axis.
- Similar to option A, but the reflection would mean altering the result of the rotation, likely changing the y-coordinates afterward.

**C**: A rotation 180 degrees clockwise about the origin, followed by reflection across the y-axis.
- The 180-degree rotation would again invert coordinates and could work with the subsequent reflection altering one axis.

**D**: A rotation 180 degrees clockwise about the origin, followed by a translation 2 units to the left.
- This would produce a consistent transformation, but like before, the translation afterward is a key factor to identify its effectiveness.

**Conclusion for Part B:**
The transformations that involve a combination of rotations are generally effective for variety in shapes, so options A and B are likely preferable depending on the final arrangements of figure 3.

If the actual figures were visible, it would be clearer which options are accurate for transformation sequences.