COMPOSITION OF

TRANSFORMATIONS

Warm Up

Determine the coordinates of the image of P(4, –7) under each transformation.

1. a translation 3 units left and 1 unit up

2. a rotation of 90° about the origin

(1, –6)

(7, 4)

3. a reflection across the y-axis

(–4, –7)

You drew reflections, translations, and rotations.

• Draw glide reflections and other compositions of isometries in the coordinate plane.

• Draw compositions of reflections in parallel and intersecting lines.

Composite Photograph

Composite photographs are made by superimposing one or more photographs.

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MorphingMorphing is a popular special effect in movies.

It changes one image into another.

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DefinitionWhen a transformation is applied to a figure,

and then another transformation is applied to its image, the result is called a composition of the transformations.

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Find a single transformation for a 75° counterclockwise rotation with center

(2,1) followed by a 38° counterclockwise rotation with center (2,1)

113° counterclockwise rotation with center (2,1)

75°38°

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Find a single transformation equivalent to a translation with vector <−2, 7> followed by a

translation with vector <9, 3>.

Translation with vector <7, 10>

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Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis.

Step 1 translation along 5, 0 (x, y) → (x + 5, y)B(–3, 4) → B'(2, 4) G(–1, 3) → G'(4, 3) S(–4, 2) → S'(1, 2)T(–1, 1) → T'(4, 1)

Step 2 reflection in the x-axis (x, y) → (x, –y)B'(2, 4) → B''(2, –4) G'(4, 3) → G''(4, –3) S'(1, 2) → S''(1, –2)T'(4, 1) → T''(4, –1)

A. R'

B. S'

C. T'

D. U'

Quadrilateral RSTU has vertices R(1, –1), S(4, –2), T(3, –4), and U(1, –3). Graph RSTU and its image after a translation along –4, 1 and a reflection in the x-axis. Which point is located at (–3, 0)?

Definition

An isometry is a transformation that preserves distance.

Translations, reflections and rotations are isometries.

The composition of two or more isometries – reflections, translations, or rotations results in an image that is congruent to its preimage.

Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane.

Two translations equal

One translation14

Reflections over two parallel lines equals

One translation16

Copy and reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''.

Step 1 Reflect EFGH in line p.Step 2 Reflect E'F'G'H' in line q.

Answer: EFGH is transformed onto E''F''G''H'' by a translation down a distance that is twice the distance between lines p and q.

Reflections over two intersection lines equals

One rotation18

Graph Other Compositions of IsometriesΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin.Step 1 translation along –1 , 5 (x, y) → (x + (–1), y + 5)T(2, –1) → T'(1, 4) U(5, –2) → U'(4, 3) V(3, –4) → V'(2, 1)

Step 2 rotation 180 about the origin (x, y) → (–x, –y)T'(1, 4) → T''(–1, –4) U'(4, 3) → U''(–4, –3) V'(2, 1) → V''(–2, –1)

A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

Step 1 A brick is copied and translated to the right one brick length.

Step 2The brick is then rotated 90° counterclockwise about point M, given here.

The new brick is in place.

Step 3

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The symbol for a composition of transformations is an open circle. 

The notation

is read as a reflection in the x-axis following a translation of (x+3, y+4).  Be careful!!!  The process is done in reverse!!

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You may see various notations which represent a composition of transformations:

could also be indicated by

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• Rotation of d degrees of the point (x,y): Rd(x,y)

• Translation of vector of the point (x,y): Ta,b(x,y)

• Reflexion across the x-axis of the point (x,y): rx-axis(x,y)

• Reflexion across the y-axis of the point (x,y): ry-axis(x,y)

Symbology

Your Turn to Try a Few