Composition (Sequences) of Transformations

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When two or more transformations are combined to form a new transformation, the result is called a composition of transformations, or a sequence of transformations. In a composition, one transformation produces an image upon which the other transformation is then performed.
In Algebra 2, you will see "composition of functions" which will work in this same manner.

The symbol for a composition of transformations (or functions) is an open circle.
A notation such as compTis read as:
"a translation of (x, y) → (x + 1, y + 5) after a reflection in the line y = x".
You may also see the notation written as combonot().

beware This process must be done from right to left (greenarrow)!!comppic
Composition of transformations is not commutative.
As the graphs below show, if the transformation is read from left to right,
the result will NOT be the same as reading from right to left.


Let's look at some special situations involving combinations:

bullet In certain cases, a combination of transformations may be renamed by a single transformation.

Example: rotatenew2

The combination of a line reflection in the y-axis, followed by a line reflection in the x-axis, can be renamed as a single transformation of a rotation of 180ยบ (in the origin).
It is not possible to rename all compositions of transformations with one transformation, however:
Any translation or rotation can be expressed as the composition of two reflections.


bullet A composition of reflections over two parallel lines is equivalent to a translation.
(May also be over any even number of parallel lines.)

Example:   Given a || b, and pre-image ΔABC, where parallel lines are vertical. combop2

Look carefully in this situation to see which of the parallel lines will be the first line of reflection. Do not assume the parallel line nearest the pre-image (as in this example) will always be used first.

It can be seen from the diagram, that ΔA''B''C'' could also be a horizontal translation of ΔABC. The horizontal distance of the translation will be twice the width between the vertical parallel lines.   combop3

The parallel lines may be vertical (as seen in this example), horizontal or slanted.


bullet The composition of reflections over two intersecting lines is equivalent to a rotation.
The center of rotation is the intersection point of the lines.

Example:   Given two lines, a and b, intersecting at point P, and pre-image ΔABC. combop2


Be careful to observe which of the parallel lines will be the first line of reflection. Do not assume the parallel line nearest the pre-image (as in this example) will always be used first.

The double reflections are equivalent to a rotation of the pre-image about point P of an angle of rotation which is twice the angle formed between the intersecting lines (theta).combrot2theta
Remember that, by convention, the angles are read in a counterclockwise direction. You may also apply this rule to negative angles (clockwise).


bullet A glide reflection is the composition of a reflection and a translation, where the line of reflection, m, is parallel to the directional vector line, v, of the translation.

Example: glidenot

A glide reflection is commutative. Reversing the direction of the composition will not affect the outcome.

Footprints are an example of several glide reflections. In the diagram at the left, you are seeing the original "step" on the left foot, followed by the "step" on the right foot, which is the "result" of the glide reflection. When compared to the diagram of the triangles, shown above, you are not seeing ΔA'B'C' (reflection) in the footprints.


bullet The composition of two rotations from the same center, is a rotation whose degree of rotation equals the sum of the degree rotations of the two initial rotations.

Example: rotrotb

Note that CP = CP' = CP'', as they are radii of circle C.



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