CMClearMathAcademy

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. In Geometry Foundations, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

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).
  4. Connect the calculation back to Transformations on the Coordinate Plane so the method, not just the arithmetic, is clear.

Answer: (7, 3)

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response). Counts include repeated prompts and variants.

Practice problems

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

Show solution
  1. x: 2 - 3 = -1.
  2. y: 5 - 1 = 4.

Answer: (-1, 4)

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

Show solution
  1. Across x-axis: keep x, negate y.
  2. Result: (3, -4).

Answer: (3, -4)

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

Show solution
  1. Across y-axis: negate x, keep y.
  2. Result: (2, 5).

Answer: (2, 5)

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

Show solution
  1. Rule: (x, y) -> (-y, x).
  2. (2, 0) -> (0, 2).

Answer: (0, 2)

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

Show solution
  1. Rule: (x, y) -> (-x, -y).
  2. (3, 0) -> (-3, 0).

Answer: (-3, 0)

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