Transformations

Review There are four types of transformations Translations Reflections Rotations Dilations

Translations A translation is a transformation that moves the points of an image the same distance and in the same direction—without rotating, resizing or anything else, just moving (or sliding)

Example (3,6) (6,5) (5,1) (3,-2) (6,-3) (5,-7)

Example 2

Reflections A reflection is a transformation when a figure is flipped over a line of reflection (its like looking in a mirror).

Common reflections

Horizontal reflection Reflecting over the 𝑦−𝑎𝑥𝑖𝑠 (𝑥,𝑦)→(−𝑥,𝑦) Reflecting across 𝑥=𝑎, when 𝑎 is a given number

Vertical reflection When reflecting over the 𝑥−𝑎𝑥𝑖𝑠 (𝑥,𝑦)→(𝑥,−𝑦) Reflecting across 𝑦=𝑎, when 𝑎 is a given number

Example 3

Example 4

Example 1 Use the coordinate mapping (x, y) → (x + 8, y + 3) to translate ΔSAM to create ΔS’A’M’.

6.7 Similarity Transformations Dilations Objectives: To use dilations to create similar figures To perform dilations in the coordinate plane using coordinate notation

Dilations A dilation is a type of transformation that enlarges or reduces a figure. The dilation is described by a scale factor and a center of dilation.

Dilations The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage.

Example 2 What happens to any point (x, y) under a dilation centered at the origin with a scale factor of k?

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor. Enlargement: k > 1.

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor. Reduction: 0 < k < 1.

Example 3 Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation. Is this a reduction or an enlargement?

Example 4 A graph shows PQR with vertices P(2, 4), Q(8, 6), and R(6, 2), and segment ST with endpoints S(5, 10) and T(15, 5). At what coordinate would vertex U be placed to create ΔSUT, a triangle similar to ΔPQR?

Example 5 Figure J’K’L’M’N’ is a dilation of figure JKLMN. Find the coordinates of J’ and M’.

Dilations not about the origin