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Transformations on the Coordinate Plane

A free Pre-Algebra lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

Three rigid motions move figures without changing their size or shape. Translation slides by adding to each coordinate. Reflection flips across an axis (negates the corresponding coordinate). Rotation around the origin sends (x, y) to predictable new positions for 90, 180, and 270 degree turns.

What you'll learn

Why it matters: Video games, robotics, architecture, and animation all use coordinate transformations to move and orient objects without distorting them.

Worked example

Problem. Translate (3, 1) by 4 right and 2 up.

  1. Add 4 to the x-coordinate: 3 + 4 = 7.
  2. Add 2 to the y-coordinate: 1 + 2 = 3.
  3. New point: (7, 3).

Answer: (7, 3)

Practice problems

1. Translate (2, 5) by 3 left and 1 down. Enter as (x, y).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Translate (2, 5) by 3 left and 1 down. Enter as (x, y).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. x: 2 - 3 = -1.
  4. y: 5 - 1 = 4.
  5. Check the result by substituting or estimating: the response should match (-1, 4) and make sense in the original problem.

Answer: (-1, 4)

2. Reflect (3, 4) across the x-axis. Enter as (x, y).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Reflect (3, 4) across the x-axis. Enter as (x, y).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Across x-axis: keep x, negate y.
  4. Result: (3, -4).
  5. Check the result by substituting or estimating: the response should match (3, -4) and make sense in the original problem.

Answer: (3, -4)

3. Reflect (-2, 5) across the y-axis. Enter as (x, y).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Reflect (-2, 5) across the y-axis. Enter as (x, y).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Across y-axis: negate x, keep y.
  4. Result: (2, 5).
  5. Check the result by substituting or estimating: the response should match (2, 5) and make sense in the original problem.

Answer: (2, 5)

4. Rotate (2, 0) by 90 degrees counterclockwise around the origin. Enter as (x, y).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Rotate (2, 0) by 90 degrees counterclockwise around the origin. Enter as (x, y).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Rule: (x, y) -> (-y, x).
  4. (2, 0) -> (0, 2).
  5. Check the result by substituting or estimating: the response should match (0, 2) and make sense in the original problem.

Answer: (0, 2)

5. Rotate (3, 0) by 180 degrees around the origin.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Rotate (3, 0) by 180 degrees around the origin.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Rule: (x, y) -> (-x, -y).
  4. (3, 0) -> (-3, 0).
  5. Check the result by substituting or estimating: the response should match (-3, 0) and make sense in the original problem.

Answer: (-3, 0)

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