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Difficulty: MediumTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, point PP is rotated 90∘90^\circ clockwise about the origin, and then translated 33 units to the left and 44 units up to form the image point P′(1,2)P'(1, 2). What are the coordinates of the pre-image point PP?

  1. A
    (−1,3)(-1, 3)
  2. B
    (−6,−2)(-6, -2)
  3. (2,4)(2, 4)Answer
  4. D
    (−2,−4)(-2, -4)
  5. E
    (1,−3)(1, -3)

Answer

The coordinates of the pre-image point PP are (2,4)(2, 4).
To find the coordinates of the pre-image point, we must undo the transformations in reverse order. First, we undo the translation of 33 units left and 44 units up by translating the image point (1,2)(1, 2) by 33 units right and 44 units down, yielding the intermediate point (4,−2)(4, -2). Next, we undo the 90∘90^\circ clockwise rotation by rotating this intermediate point 90∘90^\circ counterclockwise about the origin. Using the rotation rule (x,y)→(−y,x)(x, y) \rightarrow (-y, x) on (4,−2)(4, -2) gives the pre-image point (2,4)(2, 4).

Step-by-Step Solution

1
Identify the sequence of forward transformations and set up the path to work backward from the final image point P′(1,2)P'(1, 2) to the pre-image point PP.
The forward sequence is: P→Rotate 90∘ clockwisePmid→Translate left 3, up 4P′(1,2)P \xrightarrow{\text{Rotate } 90^\circ \text{ clockwise}} P_{\text{mid}} \xrightarrow{\text{Translate left 3, up 4}} P'(1, 2). To find PP, we must apply the inverse transformations in reverse order: P′(1,2)→Translate right 3, down 4Pmid→Rotate 90∘ counterclockwisePP'(1, 2) \xrightarrow{\text{Translate right 3, down 4}} P_{\text{mid}} \xrightarrow{\text{Rotate } 90^\circ \text{ counterclockwise}} P.
Working backward with inverse operations is the standard mathematical procedure to retrieve a pre-image from its transformed image.
2
Apply the inverse translation to the image point P′(1,2)P'(1, 2) to find the coordinates of the intermediate point PmidP_{\text{mid}}.
To undo a translation of 33 units left and 44 units up, translate 33 units right and 44 units down: x=1+3=4x = 1 + 3 = 4, and y=2−4=−2y = 2 - 4 = -2. Thus, Pmid=(4,−2)P_{\text{mid}} = (4, -2).
Applying the opposite direction and magnitude to the coordinates reverses the translation effect.
3
Apply the inverse rotation to the intermediate point Pmid(4,−2)P_{\text{mid}}(4, -2) to find the coordinates of the pre-image point PP.
The inverse of a 90∘90^\circ clockwise rotation is a 90∘90^\circ counterclockwise rotation about the origin. The rule for a 90∘90^\circ counterclockwise rotation is (x,y)→(−y,x)(x, y) \rightarrow (-y, x). Applying this to (4,−2)(4, -2) gives P(−(−2),4)=P(2,4)P(-(-2), 4) = P(2, 4).
Rotating 90∘90^\circ counterclockwise reverses the initial 90∘90^\circ clockwise rotation, returning the point to its original pre-image coordinates.

Key Concept

Determining the coordinates of a pre-image by reversing a composite transformation in the coordinate plane.
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Topics

Transformations in the Coordinate Plane

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