Sequences of Transformations
A sequence of transformations maps a preimage to an image by applying two or more transformations in order. The same final result can often be achieved through multiple different sequences. For triangle PQR with vertices P(2,5), Q(4,5), R(2,2) mapping to P(-2,-5), Q(-4,-5), R(-2,-2): this can be achieved by reflecting across the y-axis then the x-axis, or across the x-axis then the y-axis — both equivalent to a 180-degree rotation about the origin. This flexibility, from enVision Mathematics, Grade 8, Chapter 6, is a key insight in 8th grade geometry.
Key Concepts
A sequence of transformations, or a composition, maps a preimage figure $F$ to a final image figure $F''$. This can be represented as $F'' = (T 2 \circ T 1)(F)$, where transformation $T 1$ is applied first, followed by $T 2$. For any given preimage and image, there can be multiple different sequences of transformations that produce the same result.
Common Questions
What is a sequence of transformations?
A sequence of transformations applies two or more individual transformations (translations, reflections, rotations) in order to map a preimage to a final image.
Can two different transformation sequences produce the same result?
Yes. For example, reflecting a figure across the y-axis then the x-axis gives the same result as reflecting across the x-axis then the y-axis. Both are equivalent to a 180-degree rotation.
How do I describe a sequence of transformations?
Identify the specific transformations needed in order. A useful strategy is to first find a translation that maps one vertex to its image, then determine the rotation or reflection needed to align the rest of the figure.
How does orientation help identify if a reflection is needed?
If the vertices of the image are in the opposite order (clockwise instead of counterclockwise compared to the preimage), a reflection must be part of the sequence.
What does applying two reflections across perpendicular axes produce?
Reflecting across the y-axis then the x-axis (or vice versa) is equivalent to a 180-degree rotation about the origin.
When do 8th graders learn sequences of transformations?
Chapter 6 of enVision Mathematics, Grade 8 covers this in the Congruence and Similarity unit.