Obj. 30 Reflections and Translations
•
0 likes•2,054 views
The document describes reflections and translations, which are types of transformations in geometry. It defines reflections as transformations across a line where corresponding points are an equal distance from the line. Translations are defined as transformations where all points of a figure are moved the same distance in the same direction, making them an isometry. Examples of reflecting and translating points and figures are provided to illustrate these concepts.
Report
Share
Obj. 30 Reflections and Translations
- 1. Obj. 30 Reflections and Translations The student is able to (I can): • Identify and draw reflections • Identify and draw translations
- 2. transformation A change in the position, size, or shape of a figure. A A´ B C B´ C´ preimage The original figure. image The figure after the transformation. isometry A transformation that only changes the position of the figure.
- 3. A´ A mapping B B´ C C´ Note: We use “arrow notation” to describe a transformation and primes (´) to label the image. This process is called mapping mapping. A is mapped to A´ (A → A′) B is mapped to B´ (B → B′) C is mapped to C´ (C → C′) ∆ABC is mapped to ∆A´B´C´ (∆ABC → ∆A′B′C′)
- 4. reflection A transformation across a line; each point and its image are the same distance from the line. P(x, y) → P′(x, − y) P(x, y) → P′(− x, y) P(x, y) → P′(y, x) Across the x-axis Across the y-axis Across the line y=x P´(—x, y) y • P(x, y) • P´(y, x) x 0 • P´(x, —y)
- 5. Examples Reflect the given vertices across the line. 1. L(-2, 0), H(-1, 4), S(3, 2); y-axis y H• • • S x L 2. M(-3, 3), A(2, 3), T(2, -1), H(-3, -1); y=x M • • • • H y=x A T
- 6. Examples Reflect the given vertices across the line. 1. L(-2, 0), H(-1, 4), S(3, 2); y-axis y H• • H´ S´ • • • • L L´(2, 0) H´(1, 4) S´(-3, 2) S L´ x 2. M(-3, 3), A(2, 3), T(2, -1), H(-3, -1); y=x M • • A T´ • • • H y=x • A´ • H´ T • M´ M´(3, -3) A´(3, 2) T´(-1, 2) H´(-1, -3)
- 7. 3. Reflect the points G(-1, 5), E(0, 3), O(2, -4) a. Across the y-axis: (x, y) → (− x, y) G´(1, 5), E´(0, 3), O´(-2, -4) b. Across the x-axis: (x, y) → (x, − y) G´(-1, -5), E´(0, -3), O´(2, 4) c. Across the line y=x: (x, y) → (y, x) G´(5, -1), E´(3, 0), O´(-4, 2)
- 8. translation A transformation where all the points of a figure are moved the same distance in the same direction. It is an isometry.
- 9. Examples What are the coordinates of the translated points? 1. L(-1, 6) 5 units to the right and 4 units down. L´(4, 2) 2. R(0, 8) 2 units to the left and 5 units up. R´(-2, 13) 3. Y(7, -3) 4 units to the left and 3 units down. Y´(3, -6)
- 10. vector A quantity that has both length and direction. The vector x, y lists the horizontal and vertical change from the initial point to the final point. (Notice the angle brackets instead of parentheses.) Example Translate U(7, 2) along −2, 4 U´(7 — 2, 2 + 4) U´(5, 6)
- 11. Examples Translate the figure with the given vertices along the given vector. 1. U(-3, -1), T(1, 5), A(6, -3); 4, 4 T´ A´ U´(1, 3), T´(5, 9), A´(10, 1) 2. T(-2, -4), A(-3, 0), M(1, 0), U(2, -4); −2, 4 A´ M´ U´ T´(-4, 0), A´(-5, 4), M´(-1, 4), U´(0, 0) 3. M(-3, -1), A(5, -3), V(-2, -2); 1, −3 A´ V´ M´(-2, -4), A´(6, -6), V´(-1, -5)