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Introduction to Transformations / Translations

Introduction to Transformations / Translations. By the end of this lesson, you will know…. Transformations in general: A transformation is a change in the position, size, or shape of a geometric figure. There are two types of transformations, rigid and non-rigid motions.

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Introduction to Transformations / Translations

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  1. Introduction to Transformations / Translations

  2. By the end of this lesson, you will know… • Transformations in general: • A transformation is a change in the position, size, or shape of a geometric figure. • There are two types of transformations, rigid and non-rigid motions. • The pre-image is the original figure and the image is the transformed figure. • Congruent figures have the same size and shape. • Rigid motions preserve the size and shape (or distance and angle measure) of a figure. (They are sometimes called congruence motions.) • Rigid motions include translations, reflections, and rotations. • Translations: • A translation is a transformation where a figure slides without turning. • Lines that connect the corresponding points of a pre-image and its translated image are parallel. • Corresponding segments of a pre-image and its translated image are parallel. • A translation does not change the orientation of a figure.

  3. What is a transformation? • a change in the position, size, or shape of a geometric figure • 2 types of transformations • rigid motion • translations • reflections • rotations • non-rigid motion

  4. Rigid motions • preserve the size and shape (or distance & angle measure) of a figure • they are sometimes called congruence motions since figures with the same size and shape are congruent

  5. Transformations • are functions that take points in the plane as inputs and give other points as outputs • we will discuss & use “rules” that go with these transformations

  6. With transformations, • the pre-image is the original figure (input) • the image is the transformed figure (output) • to distinguish the pre-image from the image, prime notation is used points A, B, and C are inputs points A’, B’, and C’ are outputs

  7. Function “Rule” • If we are given a pre-image and its transformed image on a coordinate plane, we can give a function rule for the horizontal and vertical change • To give a function rule, we can use…

  8. Coordinate Notation • coordinate notation is a way to write a function rule for a transformation in the coordinate plane • example: (x,y)  (x+2, y-3) (x+2, y-3) is our function rule written in coordinate notation the pre-image is moving 2 units right and 3 units down applying this rule • if (6,12) is a point on our pre-image, then (6,12) becomes (6+2,12-3), which is point (8,9) • so point (6,12) on our pre-image transformed to point (8,9) on our image

  9. Is this transformation a rigid motion? (x,y)  (x-4,y+3) pre-image points (0,0) (2,0) (0,4)

  10. Is this transformation a rigid motion? (x,y)  (-x,y) pre-image points (0,0) (2,0) (0,4)

  11. YES this transformation is a rigid motion because the size and shape of the pre-image is preserved

  12. Is this transformation a rigid motion? (x,y)  (2x,2y) pre-image points (0,0) (2,0) (0,4)

  13. Is this transformation a rigid motion? (x,y)  (2x,y) pre-image points (0,0) (2,0) (0,4)

  14. Is this transformation a rigid motion? (x,y)  (x, .5y) pre-image points (0,0) (2,0) (0,4)

  15. Translations • a translation is a transformation that slides all points of a figure the same distance in the same direction

  16. Translations • lines that connect the corresponding points of a pre-image and its translated image are parallel. • corresponding segments of a pre-image and its translated image are also parallel • it is convenient to describe translations using the language of vectors

  17. Vector • a quantity that has both direction and magnitude (size) • the initial point of a vector is the starting point • the terminal point of a vector is the ending point

  18. Component Form of Vectors • denoted by <a,b> • specifies the horizontal change a and the vertical change b from the initial point to the terminal point • a is negative if moved left • b is negative if moved down

  19. Translations and Vectors • The translation below shows a vector (in red) translating the top triangle 4 units to the right and 9 units downward.  The notation for such vector movement may be written as <4,-9>.

  20. Draw the image of the pre-image shown below under the given translation vector: <-4,2>

  21. Determining the translation vector • given a pre-image and its translated image, determine the translation vector. Give a verbal description of the translation vector.

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