490 likes | 963 Views
Transformations. A transformation is an operation that changes some aspect of the geometric figure to produce a new figure. The new figure is called the image , and the original figure is called the pre-image. C. C’. Pre-image. Image. Transformation. A. A’. B. B’.
E N D
Transformations A transformationis an operation that changes some aspect of the geometric figure to produce a new figure. The new figure is called the image, and the original figure is called the pre-image. C C’ Pre-image Image Transformation A A’ B B’
Congruence Transformations A congruence transformation, or isometry, is a type of transformation that changes the position of a figure without changing its size or shape. • In other words, in an isometry, the pre-image is congruent to the image. • There are three basic isometries…
Isometries Which of the following transformations is not an isometry?
Tessellations An interesting application of transformations is a tessellation. A tessellation is a tiling of a plane with one or more shapes with no gaps or overlaps. They can be created using transformations.
Vectors Translations are usually done with a vector, which gives a direction and distance to move our shape.
Vectors Translations are usually done with a vector, which gives a direction and distance to move our shape.
Transformation Coordinate Rules What are the new coordinates of the point (x, y) under each of the following transformations? • Translation under the vector a, b • Reflection across the x-axis Reflection across the y-axis • Reflection across the line y= x Reflection across the line y = -x • Rotation of 90° around the origin
Transformation Coordinate Rules Coordinate Notation for a Translation You can describe a translation of the point (x, y) under the vector a, b by the notation:
Transformation Coordinate Rules Coordinate Notation for a Reflection
Transformation Coordinate Rules Coordinate Notation for a Rotation
Example 1 Draw and label ABC after each of the following transformations: • Reflection across the x-axis • Reflection across the y-axis • Translation under the vector -3, 5
Example 2 What translation vector was used to translate ABC to A’B’C’? Write a coordinate rule for the translation. Vector: a, b = 10, -2 Rule: (x, y) (x + 10, y – 2)
Example 3 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.
Example 3 Draw the image of ABC after it has been rotated 90° counterclockwise around the origin.
Example 4a Does the order matter when you perform multiple transformations in a row?
Example 4b Does the order matter when you perform multiple transformations in a row?
Example 4c Does the order matter when you perform multiple transformations in a row?
Composition of Transformations Two or more transformations can be combined to make a single transformation called a composition of transformations.
Composition of Transformations When the transformations being composed are of different types (like a translation followed by a reflection), then the order of the transformations is usually important.
Glide Reflection A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflection.
Glide Reflection A special type of composition of transformations starts with a translation followed by a reflection. This is called a glide reflection.