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Difficulty: EasyTransformations in the Coordinate Plane

In the standard (x,y)(x,y) coordinate plane, the point P(3,−4)P(3, -4) is translated 55 units to the left and 22 units up to map onto point P′P'. What are the coordinates of P′P'?

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

Answer

The coordinates of point P′P' are (−2,−2)(-2, -2).
To find the coordinates of the image point P′P' after translation, apply the shifts to the coordinates of the pre-image point P(3,−4)P(3, -4). Translating 55 units left subtracts 55 from the x-coordinate: 3−5=−23 - 5 = -2. Translating 22 units up adds 22 to the y-coordinate: −4+2=−2-4 + 2 = -2. Thus, the coordinates of P′P' are (−2,−2)(-2, -2).

Step-by-Step Solution

1
Determine the effect of translating 55 units to the left on the x-coordinate.
Subtract 55 from the x-coordinate: 3−5=−23 - 5 = -2.
Horizontal translations to the left decrease the x-value.
2
Determine the effect of translating 22 units up on the y-coordinate.
Add 22 to the y-coordinate: −4+2=−2-4 + 2 = -2.
Vertical translations upward increase the y-value.
3
Combine the new coordinates to find the image point P′P'.
P′=(−2,−2)P' = (-2, -2)
The translation maps the original point P(3,−4)P(3, -4) to the new coordinates P′(−2,−2)P'(-2, -2).

Key Concept

Translating a point (x,y)(x, y) horizontally by hh units and vertically by kk units yields the image point (x+h,y+k)(x + h, y + k). Shifts to the left and down correspond to negative values for hh and kk, while shifts to the right and up correspond to positive values.
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Topics

Transformations in the Coordinate Plane

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