Question

Point P is at (-4,-5). The point was translated by the rule (x+3,y-6). Then the point was translated again by the rule (x-4,y+2). What quadrant will point P′ be in?(1 point) Responses I I II II III III IV IV Question 2 Write the rule for the composition transformation (4 points) (x -,y-​​​​​​​)​​​​​​​

Answer 1

Let's perform the translations step by step for point P at (-4, -5).

**First Translation:**
The translation rule is (x + 3, y - 6).
1. Start with P: (-4, -5)
2. Apply the translation:
- For x: -4 + 3 = -1
- For y: -5 - 6 = -11
3. After the first translation, the new point P' is (-1, -11).

**Second Translation:**
The translation rule is (x - 4, y + 2).
1. Start with P' from the first translation: (-1, -11)
2. Apply the translation:
- For x: -1 - 4 = -5
- For y: -11 + 2 = -9
3. After the second translation, the new point P'' is (-5, -9).

**Determine the Quadrant:**
- Point P'' is at (-5, -9).
- The x-coordinate is negative and the y-coordinate is also negative, which means P'' is located in **Quadrant III**.

### Responses:
1. **Quadrant**: III
2. **Composition Transformation Rule**:
- The overall rule can be determined by combining the two transformations:
- First the x-coordinate: \(x' = x + 3\) (first translation) and then \(x'' = x' - 4\), which can be simplified:
- \(x'' = (x + 3) - 4 = x - 1\)
- For the y-coordinate: \(y' = y - 6\) (first translation) and then \(y'' = y' + 2\), which can be simplified:
- \(y'' = (y - 6) + 2 = y - 4\)

Thus, the overall transformation rule is:
\[
(x, y) \rightarrow (x - 1, y - 4)
\]

This can be expressed clearly as:
\[
(x - 1, y - 4)
\]