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?

Write the rule for the composition transformation

Answer 1

To find the final position of point \( P' \) after applying the given transformations to point \( P(-4, -5) \), we can follow these steps:

1. **First Translation**: The rule is \( (x + 3, y - 6) \).

Applying this to point \( P(-4, -5) \):
\[
P_1 = (-4 + 3, -5 - 6) = (-1, -11)
\]

2. **Second Translation**: The rule is \( (x - 4, y + 2) \).

Applying this to the new point \( P_1(-1, -11) \):
\[
P' = (-1 - 4, -11 + 2) = (-5, -9)
\]

Now, we have the final position of point \( P'(-5, -9) \).

### Determining the Quadrant
The coordinates of point \( P' \) are \( (-5, -9) \).

- The \( x \)-coordinate is negative.
- The \( y \)-coordinate is negative.

Thus, point \( P'(-5, -9) \) is located in the **third quadrant**.

### Rule for the Composition Transformation
To combine the two translation rules, we can express them as a single composition transformation.

Starting from point \( (x, y) \):
1. For the first translation \( (x + 3, y - 6) \), we rewrite the new coordinates:
\[
x' = x + 3
\]
\[
y' = y - 6
\]

2. For the second translation \( (x - 4, y + 2) \), we apply this to \( (x', y') \):
\[
x'' = x' - 4 = (x + 3) - 4 = x - 1
\]
\[
y'' = y' + 2 = (y - 6) + 2 = y - 4
\]

This means the composition transformation rule from \( (x, y) \) to \( (x'', y'') \) is:
\[
(x, y) \rightarrow (x - 1, y - 4)
\]

### Final Answer
Point \( P'(-5, -9) \) is in the third quadrant, and the rule for the composition transformation is:
\[
(x, y) \rightarrow (x - 1, y - 4)
\]