9.3 Compositions of transformations

Compositions of Transformations

A composition of transformations applies two or more transformations to a figure in sequence. The central idea is that order matters: performing transformation A then B often gives a different result than B then A. This section covers how to combine translations, reflections, rotations, and dilations, and how to predict what the composed result will be.

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Composition of Transformations Concept

A composition means you take the output of one transformation and use it as the input for the next. The original figure goes through the first transformation, and then that image goes through the second.

• Notation matters. When you see something like $R \circ T$, read it right to left: apply $T$ first, then $R$. This is the same convention used in function composition from algebra.
• The four transformations you'll compose:
  • Translations slide the figure without changing orientation
  • Reflections flip the figure across a line
  • Rotations turn the figure around a fixed point
  • Dilations scale the figure larger or smaller from a center point

Because order affects the outcome, always pay close attention to which transformation is applied first.

Performing Multiple Geometric Transformations

Different combinations of transformations produce predictable results. Knowing these shortcuts lets you simplify a composition into a single equivalent transformation, which saves real time on exams.

Translations composed together

Two translations in sequence always simplify to a single translation. You find the resulting vector by adding the two translation vectors component-wise.

Reflections composed together

This is where compositions get interesting:

• Two reflections across parallel lines produce a translation. The translation direction is perpendicular to the lines, and the distance equals twice the distance between them.
• Two reflections across intersecting lines produce a rotation. The center of rotation is the point where the lines intersect, and the rotation angle equals twice the angle between the lines (measured from the first line of reflection to the second).
• Two reflections across perpendicular lines are a special case of intersecting lines: twice 90° gives a 180° rotation (half-turn) about the intersection point.

Rotations composed together

Two rotations about the same center combine into a single rotation about that center. The resulting angle is the sum of the two individual angles. Use positive values for counterclockwise and negative for clockwise.

Note: if the two rotations have different centers, the composition is not simply a single rotation with a summed angle. That situation is more complex and typically not tested at this level.

Dilations composed together

Two dilations from the same center combine into a single dilation from that center. The resulting scale factor is the product of the two individual scale factors.

Coordinates After Transformation Compositions

When you need to find the actual coordinates of a composed transformation, work through each step one at a time, applying transformations right to left (the rightmost transformation first).

Here are the coordinate rules you'll use:

Translations: $(x, y) \rightarrow (x + a,\; y + b)$, where $(a, b)$ is the translation vector

Reflections:

• Across the x-axis: $(x, y) \rightarrow (x, -y)$
• Across the y-axis: $(x, y) \rightarrow (-x, y)$
• Across $y = x$: $(x, y) \rightarrow (y, x)$
• Across $y = -x$: $(x, y) \rightarrow (-y, -x)$

Rotations (counterclockwise about the origin):

• 90°: $(x, y) \rightarrow (-y, x)$
• 180°: $(x, y) \rightarrow (-x, -y)$
• 270°: $(x, y) \rightarrow (y, -x)$

Dilations (centered at the origin): $(x, y) \rightarrow (kx, ky)$, where $k$ is the scale factor

Worked example: Apply $R_{90°} \circ r_{x\text{-axis}}$ to the point $(2, 5)$.

1. Read right to left. The rightmost transformation is $r_{x\text{-axis}}$, so reflect first.
2. Reflect $(2, 5)$ across the x-axis: $(x, y) \rightarrow (x, -y)$, giving $(2, -5)$.
3. Apply the 90° counterclockwise rotation to $(2, -5)$: $(x, y) \rightarrow (-y, x)$, giving $(5, 2)$.

The image of $(2, 5)$ under $R_{90°} \circ r_{x\text{-axis}}$ is $(5, 2)$.

Why order matters (quick check): If you reversed the order and applied $r_{x\text{-axis}} \circ R_{90°}$ to $(2, 5)$:

1. Rotate $(2, 5)$ by 90°: $(x, y) \rightarrow (-y, x)$, giving $(-5, 2)$.
2. Reflect $(-5, 2)$ across the x-axis: $(x, y) \rightarrow (x, -y)$, giving $(-5, -2)$.

You get $(-5, -2)$ instead of $(5, 2)$. Different order, different result.

Effects of Multiple Transformations

After composing transformations, you should be able to classify the overall effect on the figure.

Congruence vs. similarity:

• Translations, reflections, and rotations are all rigid motions (also called isometries). They preserve both shape and size, so the image is congruent to the original.
• Any composition of only rigid motions also produces a congruent figure.
• Dilations change size (unless the scale factor is 1 or $-1$), so if a dilation is part of the composition, the image is similar to the original but not congruent.

Useful patterns to recognize:

• Two reflections across perpendicular lines = 180° rotation about their intersection
• Two dilations with reciprocal scale factors ($k$ and $\frac{1}{k}$) from the same center cancel out, returning the figure to its original size
• A composition of a dilation and a rigid motion is called a similarity transformation. The image will always be similar to the pre-image.

Checking your work with coordinates:

• Translations shift every coordinate by the same constant amounts
• Reflections change the signs of specific coordinates (depending on the line of reflection)
• Rotations cycle and/or negate coordinates in a predictable pattern
• Dilations multiply every coordinate by the same scale factor

If your final coordinates don't follow these patterns, go back and check each step. A common mistake is applying the transformations left to right instead of right to left.