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Unit Overview

Section 12-2 Transformations - Translations SPI 32D: determine whether the plane figure has been reflected given a diagram and vice versa. Objectives: Describe translations using vectors Find translation images using matrix and vector sums.

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Unit Overview

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  1. Section 12-2 Transformations - Translations SPI 32D: determine whether the plane figure has been reflected given a diagram and vice versa • Objectives: • Describe translations using vectors • Find translation images using matrix and vector sums Unit Overview • Investigate Transformations • Reflections (this lesson) • Translations • Rotations • Composition of Transformations

  2. Translation (Slide) Isometry that maps all points of a figure the same distance in the same direction. The translation at the right maps VV’ and AA’ so that: AA’VV’ is a parallelogram if A, V, and V’ are noncollinear. AA’ = VV’ and AV = A’V’ if A, V, and V’ are collinear.

  3. Recall Matrices and Vectors Enter the Matrix x-coordinate y-coordinate • Vectors are written as: • <2, 3> • indicates direction of move for each point • Values correspond to (x, y)

  4. Write a Rule to Describe a Translation Write a rule to describe the translation PQRS → P’Q’R’S’ Take any point & its corresponding prime to find the vector. Use P(-1, -2) and its image P’(-5, -1) Find the horizontal change. -5 – (-1) = -4 Find the vertical change. -1 – (-2) = -1 The vector is: < -4, 1 > The Rule is: (x, y) → (x – 4, y + 1)

  5. Using Matrices to Find Images Use matrices to find the image of ∆MFH under the translation <4, -5>.

  6. Compositions A composition is a combination of two or more transformations. Each transformation is performed on the image of the preceding transformation. Example A game of chess where a knight is moved. The knight moves in a ‘L’ shape…. translate 3 units down and slide over 1 unit.

  7. Real-World: Using Compositions and Vectors Howard rides his bicycle 3 blocks east and 5 blocks North of a pharmacy to deliver a prescription. Then he rides 4 blocks west and 8 blocks south to make a second delivery. How many blocks is he now from the pharmacy? The vector 3, 5 represents a ride of 3 blocks east and 5 blocks north. The vector –4, –8 represents a ride of 4 blocks west and 8 blocks south Howard’s position after the second delivery is the sum of the vectors. 3, 5 + –4, –8 = 3 + (–4), 5 + (–8) = –1, –3, so Howard is 1 block south and 3 blocks west of the pharmacy.

  8. Do Now! Draw each figure on graph paper, and apply the given vector to draw the image. 1. 2.

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