Translations

Unit 9: Transformations · ~15–30 min

What a transformation is

A transformation moves every point of a figure according to a rule, producing a new figure called the image. The original figure is the preimage. If a point A maps to a new point, that new point is labeled A′ (read "A prime"). A transformation that preserves size and shape — so the image is congruent to the preimage — is called an isometry. Translations, reflections, and rotations are all isometries.

Translations

A translation slides every point of a figure the same distance in the same direction. Nothing rotates, flips, or resizes — the image is just the preimage picked up and set down somewhere else. A translation is described by a vector: how far to move horizontally and how far to move vertically.

(x, y) → (x + a, y + b)

Here a is the horizontal shift (positive means right, negative means left) and b is the vertical shift (positive means up, negative means down). Every point of the figure — and every vertex of a polygon — uses the exact same a and b.

Guided practice

• 1.Translate point B(−2, 6) using the rule (x, y) → (x + 4, y − 5).
  Show answer

  −2 + 4 = 2, and 6 − 5 = 1, so B′(2, 1).
• 2.Square WXYZ has vertex W(0, 0). After a translation, W′(−6, 3). What is the translation rule?
  Show answer

  −6 − 0 = −6 and 3 − 0 = 3, so the rule is (x, y) → (x − 6, y + 3).
• 3.Triangle GHI has vertices G(3, −1), H(5, −1), I(4, 2). Apply (x, y) → (x − 2, y + 6).
  Show answer

  G′(1, 5), H′(3, 5), I′(2, 8).
• 4.Is a translation an isometry? Explain.
  Show answer

  Yes. Every point moves the same distance in the same direction, so all distances and angles in the figure stay the same — the image is congruent to the preimage.
• 5.A figure is translated 7 units left and 0 units vertically. Write the coordinate rule, then apply it to the point (−1, 8).
  Show answer

  The rule is (x, y) → (x − 7, y). Applied to (−1, 8): (−1 − 7, 8) = (−8, 8).