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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. In Coordinate Geometry and Proof, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

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).
  4. Connect the result back to Transformations on the Coordinate Plane so the geometric relationship is explicit.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: Reflection

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