1. Transformations: Mappings and Functions Lesson 14.1 Pre-AP Geometry

2. Lesson Focus The purpose of this lesson is to introduce and define the concept of a mapping. The lesson also develops two other basic ideas, namely transformations of the plane and distance preserving mappings called isometries.

3. Basic Terms Transformation The operation that maps, or moves, a preimage onto an image. Three basic transformations are reflections, rotations, and translations. Isometry A transformation that preserves lengths. Also called a rigid transformation or congruence mapping. Geometric figures which can be related by an isometry are called congruent. The idea of isometry has many uses in art, architecture, and mechanical engineering.

4. Basic Terms Image The new figure that results from the transformation of a figure in a plane. Preimage The original figure in the transformation of a figure in a plane.

5. Basic Terms Mapping A correspondence between points. Each point P in a given set is mapped to exactly one point P’ in the same or a different set. P’ is called the image of P, and P is called the preimage of P’. Function A correspondence between sets of numbers in which each number in the first set corresponds to exactly one number in the second set.

6. Basic Terms One-to-one mapping (or function) A mapping (or function) from set A to set B in which every member of B has exactly one preimage in A.

7. Mapping Notation In algebra, the function f that shows the relationship of a value x in set A to a value y in set B is expressed as f(x) = y or f:x→y In geometry, the mapping of a point P in the preimage to the point P’ in the image is expressed as M(P)=P’ or M:P→P’

8. Theorem 14-1 An isometry maps a triangle to a congruent triangle.

9. Practice #1 Given: Function k maps every number to a number that is two less than one-third of the number. 1. Express this fact using function notation. 2. Find the image of 9. 3. Find the preimage of 16.

10. Corollary 1 An isometry maps an angle to a congruent angle.

11. Corollary 2 An isometry maps a polygon to a polygon with the same area.

12. Practice #2 Given: Mapping T maps each point (x, y) to the point (x + 2, 3y). 1. Express this fact using mapping notation. 2. Find P’ and Q’, the images of P(2, 4) and Q(-2, 6). 3. Decide whether T maps M, the midpoint of PQ to M’, the midpoint of P’Q’. 4. Decide whether PQ = P’Q’.

13. Notes By definition, an isometry preserves distance. You can think of an isometry as keeping a figure rigid. Because the figure is kept rigid, its image will be a congruent figure. Until it is proven that a given transformation maps every segment to a congruent segment, you may not claim that a transformation is an isometry. You may say that it appears to be an isometry based on your experiments with particular segments.

14. Practice #3 Given: Mapping S maps each point (x, y) to an image point (x, -2y). Also, A(-3, 1), B(-1, 3), C(4, 1), and D(2, -1). 1. Decide whether: AB = A’B’ BC = B’C’ AC = A’C’ CD = C’D’ 2. Is S an isometry? Explain.

15. Written Exercises Problem Set 14.1, p.574: # 2 - 10 (even); Handout 14-1