Composition of Transformations Math Example 4

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Example 4

medium
Describe the single transformation equivalent to two successive reflections over parallel lines x=1x = 1 and x=4x = 4.

Solution

  1. 1
    Reflecting over x=1x = 1 then x=4x = 4: the distance between the lines is 4−1=34 - 1 = 3.
  2. 2
    Two reflections over parallel lines produce a translation perpendicular to those lines, with magnitude equal to twice the distance between the lines: 2×3=62 \times 3 = 6 units.
  3. 3
    The equivalent single transformation is a translation of 66 units in the positive xx-direction.

Answer

A translation of 66 units in the positive xx-direction.
Two reflections over parallel lines always compose to a translation. The translation distance is twice the gap between the lines, directed from the first reflection line toward the second.

About Composition of Transformations

Composition of transformations applies two or more transformations in sequence to a figure, where the output of one transformation becomes the input of the next. The order matters because transformation composition is generally not commutative.

Learn more about Composition of Transformations →

More Composition of Transformations Examples

Example 1 medium

Point [formula] is first reflected over the [formula]-axis, then translated by vector [formula]. Fin

Example 2 hard

Triangle [formula] with [formula], [formula], [formula] is rotated [formula] counterclockwise about

Example 3 easy

Point [formula] is translated by [formula] and then reflected over the [formula]-axis. Find the fina

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