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Sequences (Composition) of Transformations
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When two or more transformations are combined to form a new transformation, the result is called a sequence of transformations or a composition of transformations. In a sequence, one transformation produces an image upon which the other transformation is then applied.
The image of the first transformation becomes the pre-image of the next transformation.

Example: Given quadrilateral ABCD with A(3,4), B(5,4), C(4,1) and D(2,1).
Reflect the figure over the x-axis, then translate that result by the mapping (x, y) → (x - 5, y + 2).

Reflect
Reflect then Translate

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Consider these two transformations used in a sequence:
• "a reflection in the line y = x"  
• "a translation of (x, y) → (x + 1, y + 5)"

Notation: The symbol for a sequence (or composition) of transformations is an open circle.

 compT

A notation such as compTis read as:
"a translation of (x, y) → (x + 1, y + 5) after "a reflection in the line y = x".

beware This process must be done from right to left (greenarrow)!!

Now, let's take a look at these two transformations:

Graphed as listed above:
"a reflection in the line y = x"     
followed by     
"a translation of (x, y) → (x + 1, y + 5)"

combort
Graphed in reverse order:
"a translation of (x, y) → (x + 1, y + 5)"
followed by     
"a reflection in the line y = x"     

combotr

Sequences (compositions) of transformations are
not necessarily commutative.

As the graphs above show, if a transformation is read from right to left,
the result will NOT necessarily be the same as reading from left to right.
comboNOT


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ΔDOG has vertices D(-3,1), O(-2,4), and G(2,1). Rotate the triangle 180º about the origin and then translate it along the vector . Graph this sequence to find the final position of the figure.

Rotate

Translate



Given the diagram shown below, examine the graph and then answer the associated questions. Supply coordinate mapping formulas.

Given:

a) Following the blue arrows, describe the composition of transformations that occurred.

A reflection over the y-axis, followed by a reflection over the x-axis.
(x, y) → (-x, y) followed by
(x, y) → (x, -y)

b) Following the red arrow, describe the one transformation that occurred.

A reflection in the origin.
(or a rotation of 180º about the origin)
(x, y) → (-x, -y)

c) True or False: It may be possible that a composition of transformations can be replaced by a single transformation.
True


Renaming a sequence (composition):  It is possible that a sequence (composition) of two transformations may be renamed by only one other transformational method.  For example, the sequence (composition) of a line reflection in the y-axis followed by a line reflection in the x-axis could be described as a single transformation of a reflection in the origin (as seen in Ex. 2).


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