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)

Answer 1

To determine the new position of point P after the translations, we start with the original coordinates of point P, which are \((-4,5)\).

**First Translation:**
The rule for the first translation is \((x + 3, y - 6)\).

1. Calculate the new x-coordinate:
\[
-4 + 3 = -1
\]
2. Calculate the new y-coordinate:
\[
5 - 6 = -1
\]
So after the first translation, point P becomes:
\[
P' = (-1, -1)
\]

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

1. Calculate the new x-coordinate:
\[
-1 - 4 = -5
\]
2. Calculate the new y-coordinate:
\[
-1 + 2 = 1
\]
So after the second translation, point P' becomes:
\[
P'' = (-5, 1)
\]

**Determining the Quadrant:**
The coordinates \((-5, 1)\) place point P'' in the second quadrant (II), where x is negative and y is positive.

**Answer to Question 1:**
- Point P' is in Quadrant II.

---

**Question 2: Write the rule for the composition transformation.**

To find the rule for the composition, we need to combine both transformations systematically:

1. **First translation:** \((x + 3, y - 6)\)
2. **Second translation:** For the output of the first translation, we can plug in:
\[
\text{New } x = x' - 4 \Rightarrow (x + 3) - 4 = x - 1
\]
\[
\text{New } y = y' + 2 \Rightarrow (y - 6) + 2 = y - 4
\]

Putting it all together, the composition transformation rule from point \((x, y)\) is:
\[
(x, y) \rightarrow (x - 1, y - 4)
\]

**Answer to Question 2:**
- The rule for the composition transformation is \((x - 1, y - 4)\).