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) (x -,y-​​​​​​​)​​​​​​​

Answer 1

Let's perform the translations step by step for point P at (-4, -5).

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

1. Start with P: (-4, -5)
2. Apply the translation:
   • For x: -4 + 3 = -1
   • For y: -5 - 6 = -11
3. After the first translation, the new point P' is (-1, -11).

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

1. Start with P' from the first translation: (-1, -11)
2. Apply the translation:
   • For x: -1 - 4 = -5
   • For y: -11 + 2 = -9
3. After the second translation, the new point P'' is (-5, -9).

Determine the Quadrant:

• Point P'' is at (-5, -9).
• The x-coordinate is negative and the y-coordinate is also negative, which means P'' is located in Quadrant III.

Responses:

1. Quadrant: III
2. Composition Transformation Rule:
   • The overall rule can be determined by combining the two transformations:
   • First the x-coordinate: \(x' = x + 3\) (first translation) and then \(x'' = x' - 4\), which can be simplified:
     • \(x'' = (x + 3) - 4 = x - 1\)
   • For the y-coordinate: \(y' = y - 6\) (first translation) and then \(y'' = y' + 2\), which can be simplified:
     • \(y'' = (y - 6) + 2 = y - 4\)

Thus, the overall transformation rule is: \[ (x, y) \rightarrow (x - 1, y - 4) \]

This can be expressed clearly as: \[ (x - 1, y - 4) \]