Question

A) 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)
• |I_
• IV
B) Write the rule for the composition transformation (4 points)
(x-[,y-|)

Answer 1

To find the final position of point P after the two translations, let's go through the transformations step by step.

**Step 1: Initial Point P**

Point P is at (-4, -5).

**Step 2: First Translation (x + 3, y - 6)**

Using the first translation rule:
- New x-coordinate: -4 + 3 = -1
- New y-coordinate: -5 - 6 = -11

So after the first translation, point P' is at (-1, -11).

**Step 3: Second Translation (x - 4, y + 2)**

Using the second translation rule:
- New x-coordinate: -1 - 4 = -5
- New y-coordinate: -11 + 2 = -9

So after the second translation, point P'' is at (-5, -9).

**Step 4: Determine the Quadrant**

Since point P'' is at (-5, -9):
- The x-coordinate is negative.
- The y-coordinate is negative.

This places point P'' in Quadrant III.

**Answer to A**: III

---

**B) Write the rule for the composition transformation**

Now let's find the overall transformation rule from the original point P to P''.

1. The first transformation can be expressed as:
\[
(x, y) \rightarrow (x + 3, y - 6)
\]

2. The second transformation can be expressed as:
\[
(x, y) \rightarrow (x - 4, y + 2)
\]

Combining the transformations:

- For the x-coordinate:
\[
x' = (x + 3) - 4 = x - 1
\]

- For the y-coordinate:
\[
y' = (y - 6) + 2 = y - 4
\]

So the overall transformation can be expressed as:
\[
(x, y) \rightarrow (x - 1, y - 4)
\]

**Answer to B**: (x - 1, y - 4)