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Transformations on the Coordinate Plane

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

Transformations move figures while preserving or changing their position. Translations slide, reflections flip, and rotations turn around a point.

What you'll learn

Why it matters: Animators and game developers use translations, reflections, and rotations to move objects predictably on screen.

Worked example

Problem. Reflect (4, -7) across the x-axis.

  1. A reflection across the x-axis keeps x the same.
  2. The y-coordinate changes sign.
  3. (4, -7) becomes (4, 7).

Answer: (4, 7)

Practice problems

1. Translate (2, 5) right 3 and down 4.

Choices: (5, 1) · (-1, 9) · (5, 9) · (-1, 1)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Translate (2, 5) right 3 and down 4.
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Add 3 to x and subtract 4 from y.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (5, 1)

2. Reflect (-3, 8) across the x-axis.

Choices: (-3, -8) · (3, 8) · (3, -8) · (-8, -3)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Reflect (-3, 8) across the x-axis.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Keep x, change y's sign.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-3, -8)

3. Reflect (-6, -2) across the y-axis.

Choices: (6, -2) · (-6, 2) · (6, 2) · (-2, -6)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Reflect (-6, -2) across the y-axis.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Change x's sign, keep y.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (6, -2)

4. Rotate (4, 1) 180 degrees about the origin.

Choices: (-4, -1) · (-1, 4) · (1, -4) · (4, -1)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Rotate (4, 1) 180 degrees about the origin.
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A 180-degree rotation changes both signs.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-4, -1)

5. Which transformation preserves size and shape?

Choices: Reflection · Dilation by 2 · Stretch only x · Changing one side

Show solution
  1. Challenge: First identify exactly what the question is asking: Which transformation preserves size and shape?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Reflections are rigid motions.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Reflection

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