Graphing & Describing “Dilations” Chapter 6 Day 7 Graphing & Describing “Dilations”

Let’s get started! Section 1: Congruency transformations vs Similarity transformations. Section 2: Graphing a dilation. Section 3: Describing a dilation.

Congruency Transformations Congruency transformations preserve the size and shape of the original figure. The three types of congruency transformations that we have learned about are: “Slide” “Flip” “Turn”

Similarity Transformations Similarity transformations are a transformation that will produce an image that is the same shape as the pre-image but a different size. A Dilation is a similarity transformation. “Change in Size”

Today’s lesson will focus on … Dilations

Moving on… Section 1: Congruency transformations vs Similarity transformations. Section 2: Graphing a dilation. Section 3: Describing a dilation.

Graphing Dilations When a shape is dilated, its size will decrease or increase. DILATION – REDUCTION DILATION - ENLARGEMENT

R S Graphing Dilations C A L E F T O A scale factor is a multiplier that tells us the measurement of how a shape will change. When a shape is reduced, the scale is between 0 and 1. When a shape is enlarged, the scale factor is greater than 1. SCALE FACTOR < 1 SCALE FACTOR > 1 If you dilate a shape using a scale factor of 1, nothing happens. The shape is not altered in any way. So, what happens if the scale factor is 1?

R S Graphing Dilations C A L E F T O To reduce or enlarge a shape you must be given a scale factor which describes how the shape will be changed. Scale Factor = 3 To perform the dilation you will simply multiply the coordinates of each vertex by the scale factor. (x, y) (kx, ky) where k is the scale factor A(2, 1) A(3•2, 3•1) A’(6, 3) B(5, 1) B(3•5, 3•1) B’(15, 3) C(3, 6) C(3•3, 3•6) C’(9, 18)

R E D U C T I O N Guided Practice #1 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols Pre-Image A figure has vertices: 𝐉(𝟑, 𝟖) 𝐊(𝟏𝟎, 𝟔) 𝐋(𝟖, 𝟐) Graph the figure and the image of the figure after a dilation with a scale factor of ½. Image J’ ( , ) K’ ( , ) L’ ( , ) J (3, 8) --> J’ (½ • 3, ½ • 8) --> J’ (1 ½, 4) K(10, 6) --> K’ (½ • 10, ½ • 6) --> K’ (5, 3) L(8, 2) --> L’ (½ • 8, ½ • 2) --> L’ (4, 1)

E N L A R G M T Guided Practice #2 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols Pre-Image A figure has vertices: 𝐅(−𝟏, 𝟏) 𝐆(𝟏, 𝟏) 𝐇(𝟐, −𝟏) 𝐈(−𝟏, −𝟏) Graph the figure and the image of the figure after a dilation with a scale factor of 4. Image F’ ( , ) G’ ( , ) H’ , 𝐈’ ( , ) F (-1, 1) --> F’ (4 • -1, 4 • 1) --> F’ (-4, 4) G(1, 1) --> G’ (4 • 1, 4 • 1) --> G’ (4, 4) H(2, -1) --> H’(4 • 2, 4 • -1) --> H’ (8, -4) I(-1, -1) --> I’(4 • -1, 4 • -1) --> I’(-4, -4)

You Try #1 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols Pre-Image A figure has vertices: 𝐏( , ) 𝐐( , ) 𝐑( , ) 𝐒( , ) Graph the image of the figure after a dilation with a scale factor of ¼ . Image 𝐏′( , ) 𝐐′( , ) 𝐑′( , ) 𝐒′( , ) See solution on next slide.

A N S W E R You Try #1 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols See solution on next slide. Graph the image of the figure after a dilation with a scale factor of ¼ .

You Try #2 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols Pre-Image A figure has vertices: 𝐖( , ) 𝐗( , ) 𝐘( , ) 𝐙( , ) Graph the image of the figure after a dilation with a scale factor of 2 . Image 𝐖′( , ) 𝐗′( , ) 𝐘′( , ) 𝐙′( , ) See solution on next slide.

A N S W E R You Try #2 Steps for performing a Dilation Step 1: Graph and determine the coordinates of the original shape as needed Step 2: Multiply the coordinates of each vertex by the scale factor (x, y) (kx, ky) where k is the scale factor Step 3: Graph the vertices of the new image and label them with prime symbols Graph the image of the figure after a dilation with a scale factor of 2.

Last section! Section 1: Congruency transformations vs Similarity transformations. Section 2: Graphing a dilation. Section 3: Describing a dilation.

Describing a Dilation When being asked to describe a dilation that has taken place, you are usually being asked to identify a scale factor. Was the shape reduced or enlarged and by how much. If you are given two shapes and need to find the scale factor, you must know which one was the original and which one is the new image. Then you need to know the length of corresponding sides and set them up in a ratio like so:

Describing a Dilation Example: Describe the dilation that was performed below. First, determine which shape is the original and which is the image. Since the smaller one does not have “prime” marks (the little apostrophes) it must be the original one and the one with the marks is the new image. So to find the scale factor, we set up our ratio using the lengths of corresponding sides. Figure WXYZ was enlarged by a scale factor of “2” to create the image W’X’Y’Z’

Guided Practice ` Enlargement; Scale Factor = 3/2 or 1.5 Reduction; Scale Factor = 2/5

You Try ` 1) 2) Enlargement; Scale Factor = 2 Reduction; Scale Factor = 2/3

Closing Today, you learned that a dilation is a similarity transformation. When a dilation is performed on a figure, the shape of the figure will remain the same, however the size will either reduce or enlarge depending upon the scale factor used. If you are using a scale factor that is greater than 1, will the figure get bigger or smaller? What will happen to a figure that is dilated using a scale factor between 0 and 1? BIGGER It will get smaller.

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