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Zorluk: OrtaTransformations in the Coordinate Plane

The endpoint AA of segment ABAB is located at (−3,5)(-3, 5) in the standard (x,y)(x, y) coordinate plane. Segment ABAB is translated 4 units right and 2 units down, and then reflected across the line y=−xy = -x. What are the coordinates of the image of endpoint AA after these transformations?

  1. A
    (−7,7)(-7, 7)
  2. B
    (−7,−1)(-7, -1)
  3. (−3,−1)(-3, -1)Cevap
  4. D
    (3,1)(3, 1)
  5. E
    (−1,−3)(-1, -3)

Cevap

The coordinates of the image of endpoint AA are (−3,−1)(-3, -1).
Translating the point A(−3,5)A(-3, 5) 4 units right and 2 units down gives (−3+4,5−2)=(1,3)( -3 + 4, 5 - 2 ) = (1, 3). Reflecting the point (1,3)(1, 3) across the line y=−xy = -x swaps and negates the coordinates, giving (−3,−1)(-3, -1).

Adım Adım Çözüm

1
Apply the translation of 4 units right and 2 units down to the initial coordinates of point A(−3,5)A(-3, 5).
The x-coordinate changes by +4+4 and the y-coordinate changes by −2-2: (−3+4,5−2)=(1,3)(-3 + 4, 5 - 2) = (1, 3).
Translation shifts the coordinates directly by adding the horizontal change to xx and subtracting the vertical change from yy.
2
Apply the reflection across the line y=−xy = -x to the translated point (1,3)(1, 3).
Swapping and negating both coordinates of (1,3)(1, 3) yields (−3,−1)(-3, -1).
The reflection rule across the line y=−xy = -x maps any point (x,y)(x, y) to (−y,−x)(-y, -x).

Anahtar Kavram

Composite transformations in the coordinate plane involving translation and reflection across the line y=−xy = -x.
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Konular

Transformations in the Coordinate Plane

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