segment ab is reflected over y - x and then translated down 2 and right 3. If A is at (-2,3), what is the y-coordinate of A"?

Let's go step by step through the transformations:

1. **Reflecting over the line \( y = -x \):**
   - To reflect a point \( (x, y) \) over the line \( y = -x \), we swap the coordinates and change their signs. The reflection of point \( A(-2, 3) \) is:
   \[
   A' = (3, 2)
   \]

2. **Translation:**
   - Now we translate point \( A' = (3, 2) \) down by 2 units and right by 3 units. This means we subtract 2 from the y-coordinate and add 3 to the x-coordinate:
   \[
   A'' = (3 + 3, 2 - 2) = (6, 0)
   \]

The y-coordinate of \( A'' \) is \( 0 \).

Thus, the y-coordinate of \( A'' \) is **0**.

Would you like more details or have any further questions? Here are 5 related questions to explore this concept further:

1. What would be the new coordinates if the point was translated left instead of right?
2. How do you reflect a point over other lines, such as \( y = x \)?
3. How would the transformation change if the reflection was over the y-axis?
4. What is the general formula for reflecting a point over the line \( y = -x \)?
5. How would the final point differ if we translated first, then reflected?

**Tip:** When reflecting over \( y = -x \), always swap and negate both coordinates.

Learn how to find the y-coordinate of a point after reflecting segment AB over the line y = -x and translating it down 2 units and right 3 units. This problem covers key geometric transformations, including reflections and translations, suitable for students in Grades 8-10.

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