Transformations Learning Target: I will be able to translate, reflect, rotate, and dilate figures.

What is a transformation? A transformation of a geometric figure is a function, or mapping, that results in a change in the position, shape, or size of the figure. In a transformation, the original figure is the preimage and the resulting figure is the image.

Transformations Organizer On the left side of your paper, write TRANSFORMATIONS across the top. Write the definition of Transformations beneath. Transformations Definition: “A function, or mapping, that results in a change in the position, shape, or size of a figure”

Translations A translation is a transformation in which the figure maintains its shape and orientation, but is moved. The object slides to a new position, where every point of the object is moved in the same direction and the same distance.

Translations Notation Each point of ABCD is translated 4 units right and 2 units down. Which means each (𝑥,𝑦) pair in ABCD is mapped to (𝑥+4,𝑦−2). Translations may be written in the function notation 𝑇 <4,−2> (𝐴𝐵𝐶𝐷).

Transformations Organizer On the left side, below the definition of transformations, write TRANSLATION along with the definition and notation. Translation Definition: “A transformation that maps all points of a figure the same distance in the same direction, also called a slide.” Notation: 𝑇 <𝑥,𝑦> (𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒)

Transformations Organizer Cut out any shape you want, in any color, so that it will fit in one of the quadrants of the coordinate plane. Use one paper clip to attach it to quadrant II.

Problem 1: Finding the Image of a Translation Given 𝑃 2,1 , 𝑄 3,3 , and 𝑅(−1,3), what are the vertices of 𝑇 <−2,−5> (∆𝑃𝑄𝑅)? Graph the image of ∆𝑃𝑄𝑅.

Problem 2: Finding the Rule for a Translation What is a rule that describes the translation that maps 𝑃𝑄𝑅𝑆 onto 𝑃 ′ 𝑄 ′ 𝑅 ′ 𝑆′?

You Practice One: Finding the Image of a Translation Given A −1,−2 , 𝐵 0,0 , 𝐶(4, −2) and D(0,−3), what are the vertices of 𝑇 <3, 2> (𝐴𝐵𝐶𝐷)? Graph the image of 𝐴𝐵𝐶𝐷 𝑎𝑛𝑑 𝐴 ′ 𝐵 ′ 𝐶 ′ 𝐷.

Reflections A reflection is a mirror image of a figure, you can think of it as being flipped over. The preimage is reflected across a line 𝑚, called the line of reflection.

Review HOY VUX

Reflection Notation ∆𝐴𝐵𝐶 is reflected across the line 𝑚. Reflections may be written in the notation 𝑅 𝑚 ∆𝐴𝐵𝐶 , where 𝑚 is the line of reflection and ∆𝐴𝐵𝐶 is the preimage.

Transformations Organizer On the left side, below the definition of translation, write REFLECTION along with the definition and notation. Reflection Definition: “A transformation in which a figure is reflected across a line of reflection, creating a mirror image.” Notation: 𝑅 𝑙𝑖𝑛𝑒 𝑚 (𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒)

Transformations Organizer Cut out any shape you want, in any color, so that it will fit in one of the quadrants of the coordinate plane. In quadrant III, draw and label a line of reflection. It can be horizontal, vertical, or slanted. Using tape, attach your shape so that it reflects across your line of reflection.

Problem 3: Reflecting a Point Across a Line Point 𝑃 has coordinates (3,4). What are the coordinates of 𝑎 𝑅 𝑦=1 (𝑃) 𝑏 𝑅 𝑥=3 (𝑃) 𝑐 𝑅 𝑦=𝑥 (𝑃)

Problem 4: Graphing a Reflection Image Graph points 𝐴 −3,4 , 𝐵 0,1 , and 𝐶 4,2 . Graph and label 𝑅 𝑦−𝑎𝑥𝑖𝑠 (∆𝐴𝐵𝐶).

You Practice One: Graphing a Reflection Image Graph points 𝐴 −4,2 , 𝐵 −2,1 , and 𝐶 −1,4 . Graph and label 𝑅 𝑦=𝑥 (∆𝐴𝐵𝐶).

Glide Reflection A glide reflection is the composition of a translation and a reflection across a line. Glide Reflections use the notation 𝑅 𝑥=0 ∘ 𝑇 <0,−5> ∆𝑇𝐸𝑋 .

Finding a Glide Reflection Image What is 𝑅 𝑥=0 ∘ 𝑇 <0,−5> ∆𝑇𝐸𝑋 , where 𝑇 −5,2 , 𝐸 −1,3 , and 𝑋 −2,1 ?

Finding a Glide Reflection Image What is 𝑅 𝑥−𝑎𝑖𝑥 (𝑦=0) ∘ 𝑇 <0,1> ∆𝑈𝑊𝐴 , where U −5,2 , 𝑊 −1,3 , and A −2,1 ?

Rotations A rotation is a turn about a point, called the center of rotation. A figure can be rotated about its middle or about a different point, such as the origin.

Rotation Notation ∆𝑈𝑉𝑀 is rotated 𝑥° about the point 𝑄 to form ∆ 𝑈 ′ 𝑉 ′ 𝑊 ′ . Reflections may be written in the function notation 𝑟 (𝑥°,𝑄) (∆𝑈𝑉𝑊).

Transformations Organizer On the left side, below the definition of reflection, write ROTATION along with the definition and notation. Rotation Definition: “A transformation where a figure it rotated about a point, called the point of rotation.” Notation: 𝑟 (𝑥°,𝑄) (𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒)

Transformations Organizer Cut out any shape you want, in any color, so that it will fit in one of the quadrants of the coordinate plane. In quadrant I, use the brad pin to attach your shape. Label the brad pin the point of rotation.

Rotations about the Origin

Problem 5: Finding Image of a Rotation 𝑃𝑄𝑅𝑆 has vertices 𝑃 1,1 , 𝑄 3,3 , 𝑅 4,1 , and 𝑆 3,0 . What is the graph of 𝑟 90°, 𝑂 (𝑃𝑄𝑅𝑆)?

You Try One: Finding Image of a Rotation Graph 𝑟 270°, 𝑂 (𝐹𝐺𝐻𝐼) where 𝐹 −3,2 , 𝐺 −3,−1 , 𝐻 −1,−1 , 𝐼 0,1 .

Congruence Transformation A congruence transformation is a composition of rigid motions which takes figures to congruent figures.

Problem 9: Identifying Congruence Transformations In the diagram at the right, ∆𝐽𝑄𝑉≅∆𝐸𝑊𝑇. What is a congruence transformation that maps ∆𝐽𝑄𝑉 onto ∆𝐸𝑊𝑇?

Dilations A dilation is a transformation that produces an image that is the same shape, but a different size. A dilation enlarges or shrinks the original figure, which means it creates similar figures. A dilation is an enlargement if the scale factor is greater than 1, and a reduction when the scale factor is between 0 and 1.

Dilation Notation The scale factor 𝑛 is the ratio of the length of the image to the corresponding length of the preimage. Dilations may be written with the notation 𝐷 𝑛 𝐹𝐺𝐻𝐸 .

Transformations Organizer On the left side, below the definition of rotation, write DILATION along with the definition and notation. Dilation Definition: “A transformation where a figure maintains its shape but changes in size.” Notation: 𝐷 𝑛 (𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒)

Transformations Organizer Cut out any shape you want, in any color, so that it will fit in one of the quadrants of the coordinate plane. Cut out the same shape, in another color, so that it is either larger or smaller than your first shape. In quadrant IV, glue your two shapes on top of one another to show the dilation.

Problem 7: Finding a Scale Factor Is 𝐷 𝑛,𝑋 ∆𝑋𝑇𝑅 =∆ 𝑋 ′ 𝑇 ′ 𝑅′ an enlargement or a reduction? What is the scale factor 𝑛 of the dilation?

Problem 8: Finding a Dilation Image What are the coordinates of the vertices of 𝐷 2 (∆𝑃𝑍𝐺)?