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 translations, we will perform each translation step by step.

1. Initial Point P: P(-4, 5)

2. First Translation: Apply the rule (x + 3, y - 6)

   • New x-coordinate: -4 + 3 = -1
   • New y-coordinate: 5 - 6 = -1
   • New point after the first translation: P'(-1, -1)

3. Second Translation: Apply the rule (x - 4, y + 2)

   • New x-coordinate: -1 - 4 = -5
   • New y-coordinate: -1 + 2 = 1
   • New point after the second translation: P''(-5, 1)

Now, we need to determine which quadrant this new point P''(-5, 1) is in.

• The x-coordinate is -5 (negative), and the y-coordinate is 1 (positive).
• A point with a negative x-coordinate and a positive y-coordinate lies in Quadrant II.

Answer to Question 1: II

For Question 2, we need to write the rule for the composition transformation. We will combine the transformations step by step.

1. First translation: (x + 3, y - 6)
2. Second translation: (x - 4, y + 2)

To get the overall transformation:

• Start with (x, y):

  1. Apply the first transformation:

     • x becomes \( x + 3 \)
     • y becomes \( y - 6 \)

  2. Now, take that result and apply the second transformation:

     • For x: So we take \( (x + 3) - 4 = x - 1 \)
     • For y: So we take \( (y - 6) + 2 = y - 4 \)

Thus, the overall transformation rule can be written as: (x - 1, y - 4)

Answer to Question 2: (x - 1, y - 4)