9-2 Translations

1. 9-2 Translations You found the magnitude and direction of vectors. • Draw translations. • Draw translations in the coordinate plane.

2. Definition A translation is a transformation that moves all the points in a plane a fixed distance in a given direction (slide). The arrow shows the direction of the translation.

3. Definition B Terminal point or tip A Initial point or tail A vector can be represented as a “directed” line segment, useful in describing paths. A vector has both direction and magnitude (length).

4. Direction and Length From the school entrance, I went three blocks north. The distance (magnitude) is: Three blocks The direction is: North

5. Direction and Magnitude The magnitude of AB is the distance between A and B. The direction of a vector is measured counterclockwise from the horizonal (positive x-axis).

6. B B N 60° A 45° E W A S

7. Drawing Vectors Draw vector YZ with direction of 45° and length of 10 cm. Z 10 cm • Draw a horizontal dotted line • Use a protractor to draw 45° • Use a ruler to draw 10 cm • Label the points 45° Y

8. Translation vector Since vectors have a distance and a direction, they are often used to describe translations. The vector shows the direction of the translation and its length gives the distance each point travels. To measure direction, add a horizontal dotted line and measure counterclockwise

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10. Step 1 Draw a line through each vertex parallel to vector . Step 2 Measure the length of vector . Locate point G'by marking off this distance along the line through vertex G, starting at G and in the same direction as the vector. Draw a Translation Copy the figure and given translation vector. Then draw the translation of the figure along the translation vector. Answer: Step 3 Repeat Step 2 to locate points H', I', and J'to form the translated image.

11. A.B. C.D. Which of the following shows the translation of ΔABC along the translation vector?

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13. Translations in the Coordinate Plane A. Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector –3, 2. Answer: The vector indicates a translation 3 units left and 2 units up. (x, y)→ (x – 3, y + 2) T(–1, –4)→ (–4, –2) U(6, 2)→ (3, 4) V(5, –5)→ (2, –3)

14. B. Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector –5, –1. The vector indicates a translation 5 units left and 1 unit down. (x, y)→ (x – 5, y – 1) P(1, 0)→ (–4, –1) E(2, 2)→ (–3, 1) N(4, 1)→ (–1, 0) T(4, –1)→ (–1, –2) A(2, –2)→ (–3, –3) Answer:

15. Describing Translations A. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 2 to position 3 in function notation and in words. The raindrop in position 2 is (1, 2). In position 3, this point moves to (–1, –1). Use the translation function (x, y) → (x + a, y + b) to write and solve equations to find a and b. (1 + a, 2 + b) or (–1, –1) 1 + a = –1 2 + b = –1 a = –2 b = –3 Answer: function notation: (x, y) → (x – 2, y – 3) So, the raindrop is translated 2 units left and 3 units down from position 2 to 3.

16. Answer: translation vector: B. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 3 to position 4 using a translation vector. (–1 + a, –1 + b) or (–1, –4) –1 + a = –1 –1 + b = –4 a = 0 b = –3

17. B. The graph shows repeated translations that result in the animation of the soccer ball. Describe the translation of the soccer ball from position 3 to position 4 using a translation vector. A.–2, –2 B.–2, 2 C.2, –2 D.2, 2

18. 9-2 Assignment Page 627, 10-14 even, 20, 21, 26, 27