Compositions of congruence transformations: graph the image (Tutorial Only)

Learning Objectives Define a composition of transformations and identify its components. Graph the image of a figure after a sequence of two or more congruence transformations. Apply the coordinate rules for reflections, translations, and rotations in a specific sequence. Use prime notation (P -> P' -> P'') to track a point through multiple transformations. Describe a single transformation that is equivalent to a given composition, such as two reflections over parallel lines. Verify that a composition of congruence transformations results in an image that is congruent to the preimage. Ever wonder how animators make a character walk and turn in a video game? 🤖 They're using a sequence of moves, just like we will today! This tutorial will teach you...

TermDefinitionExample Composition of TransformationsThe process of applying two or more transformations in a specific order to a figure. The image of the first transformation becomes the preimage for the second transformation.A translation of 2 units right, followed by a reflection across the x-axis. Congruence Transformation (Isometry)A transformation in which the resulting image is congruent to the original preimage. The size, shape, and angle measures are preserved. Translations, reflections, and rotations are all congruence transformations.Reflecting a triangle across the y-axis produces a new triangle with the exact same side lengths and angle measures. Preimage and ImageThe preimage is the original figure before any transformations are applied. The image is the new figure that resul...

Composition Notation (T_2 \circ T_1)(P) = T_2(T_1(P)) This notation represents a composition of transformations. The transformation on the right (T_1) is applied first to the point P. Then, the transformation on the left (T_2) is applied to the result of the first transformation. Translation Rule (x, y) \rightarrow (x + a, y + b) This rule translates a point 'a' units horizontally and 'b' units vertically. If 'a' is positive, the shift is right; if negative, left. If 'b' is positive, the shift is up; if negative, down. Reflection Rules Across x-axis: (x, y) \rightarrow (x, -y) \quad | \quad Across y-axis: (x, y) \rightarrow (-x, y) \quad | \quad Across y = x: (x, y) \rightarrow (y, x) These are the coordinate rules for reflecti...