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 ′

Question

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)