https://file.aiquickdraw.com/mathbot/file/1727642072284671vxfe9.jpg

Let's break down the problem in the image step by step.

### Given Points:
- \( E(5, -6) \)
- \( L(-5, -5) \)
- \( H(6, -1) \)

### First Transformation:
The first transformation is:
\[
(x, y) \rightarrow (x - 3, y - 4)
\]

This means for each point, you subtract 3 from the x-coordinate and subtract 4 from the y-coordinate.

#### Transformations for each point:
1. **E'**:
   \[
   E'(x, y) = (5 - 3, -6 - 4) = (2, -10)
   \]
   
2. **L'**:
   \[
   L'(x, y) = (-5 - 3, -5 - 4) = (-8, -9)
   \]
   
3. **H'**:
   \[
   H'(x, y) = (6 - 3, -1 - 4) = (3, -5)
   \]

### Second Transformation:
The second transformation is a reflection over the line \( x = 0 \) (the y-axis). For this reflection, you negate the x-coordinate while keeping the y-coordinate the same.

#### Reflected points:
1. **E''**:
   \[
   E''(x, y) = (-2, -10)
   \]
   
2. **L''**:
   \[
   L''(x, y) = (8, -9)
   \]
   
3. **H''**:
   \[
   H''(x, y) = (-3, -5)
   \]

### Summary of Results:
- \( E' = (2, -10) \)
- \( L' = (-8, -9) \)
- \( H' = (3, -5) \)
- \( E'' = (-2, -10) \)
- \( L'' = (8, -9) \)
- \( H'' = (-3, -5) \)

Would you like more details or have any questions about this? 

Here are 5 related questions to consider:
1. What other transformations could be applied to the points in this triangle?
2. How would the points change if reflected over the line \( y = 0 \)?
3. What effect does reflecting over the origin have on these points?
4. How would scaling the points by a factor of 2 impact their coordinates?
5. What would happen to the triangle if we translated the points by \( (x+2, y+3) \)?

**Tip**: When reflecting over the y-axis, always flip the sign of the x-coordinate while leaving the y-coordinate unchanged!

Graph the points after the given transformations and find the location of each point after the transformations: (x, y) → (x - 3, y - 4) and reflection over the line x = 0.

This problem involves finding the new coordinates of points after translating by (x - 3, y - 4) and reflecting over the y-axis (x = 0). Suitable for students in Grades 8-10, covering key concepts in coordinate geometry, including translations and reflections.

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