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

To determine the new position of point P after the two translations, we’ll perform each translation step by step.

1. **Initial position**:
\[
P(-4, 5)
\]

2. **First translation**: \( (x + 3, y - 6) \)
\[
P' = (-4 + 3, 5 - 6) = (-1, -1)
\]

3. **Second translation**: \( (x - 4, y + 2) \)
\[
P'' = (-1 - 4, -1 + 2) = (-5, 1)
\]

Now we have the new coordinates for point P after both transformations:
\[
P''(-5, 1)
\]

To determine the quadrant:
- The x-coordinate is -5 (negative).
- The y-coordinate is 1 (positive).

Based on these coordinates:
- If x < 0 and y > 0, the point is in **Quadrant II**.

### Answer to Question 1:
**Quadrant II**

### Answer to Question 2:
To write the rule for the composition of the transformations:
1. Combine both transformations mathematically:
- Start with \( P(x, y) \), translate:
\[
(x + 3, y - 6) \rightarrow P'(x + 3, y - 6)
\]
- Then apply the second translation:
\[
P'(x + 3 - 4, y - 6 + 2) = P'(x - 1, y - 4)
\]

- The complete transformation can be expressed as:
\[
(x - 1, y - 4)
\]

### Final answer to Question 2:
**(x - 1, y - 4)**