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Difficulty: MediumTransformations in the Coordinate Plane

A triangle has a vertex at P(−4,3)P(-4, 3) in the standard (x,y)(x, y) coordinate plane. If the triangle is reflected across the line y=−xy = -x and then translated 33 units to the right and 44 units down, what are the coordinates of the image of vertex PP after both transformations?

  1. (0,0)(0, 0)Answer
  2. B
    (6,−8)(6, -8)
  3. C
    (−6,8)(-6, 8)
  4. D
    (7,−1)(7, -1)
  5. E
    (−1,−7)(-1, -7)

Answer

(0,0)(0, 0)
The coordinate (0,0)(0, 0) is correct because reflecting the point P(−4,3)P(-4, 3) across the line y=−xy = -x swaps and negates the coordinates, transforming it to (−3,4)(-3, 4). Then, translating this point 33 units to the right and 44 units down is calculated as (−3+3,4−4)(-3 + 3, 4 - 4), which simplifies to (0,0)(0, 0).

Step-by-Step Solution

1
Apply the reflection across the line y=−xy = -x to the coordinate P(−4,3)P(-4, 3).
The rule for reflection across the line y=−xy = -x is (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x). Applying this rule to P(−4,3)P(-4, 3) yields (−3,4)(-3, 4).
To find the coordinates of the vertex after the first transformation in the composite sequence.
2
Apply the translation of 33 units right and 44 units down to the intermediate point (−3,4)(-3, 4).
Translating 33 units to the right adds 33 to the xx-coordinate, and translating 44 units down subtracts 44 from the yy-coordinate: (−3+3,4−4)=(0,0)(-3 + 3, 4 - 4) = (0, 0).
To find the final position after the second transformation in the composite sequence.

Key Concept

Applying composite transformations in the coordinate plane, specifically a reflection across the line y=−xy = -x followed by a translation.
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Topics

Transformations in the Coordinate Plane

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