Unit 4 Transformations

Definitions: Transformations: It is a change that occurs that maps or moves a shape in a specific directions onto an image. These are translations, rotations, reflections, and dilations. Pre-image: The position of the shape before the change is made. Image: The position of the shape after the change is made.

Types of Transformations Reflections: These are like mirror images as seen across a line or a point. Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure. Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure. Dilations: This reduces or enlarges the figure to a similar figure.

Rigid Motion Rigid motion is otherwise known as a rigid transformation and occurs when a point or object is moved, but the size and shape remain the same.

Translations (slides) If a figure is simply moved to another location without change to its shape or direction, it is called a translation (or slide). If a point is moved “a” units to the right and “b” units up, then the translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the translated point will be at (x - a, y - b). Example: A Image A translates to image B by moving to the right 3 units and down 8 units. B A (2, 5)  B (2+3, 5-8)  B (5, -3) Vector Notation

Translation Notation Vector Notation Function Notation    

Transformation (x, y) (x + 5, y + 0) Pre-image A (-2, 4) A’ (3, 4) B B’ C C’ x Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (3, 4) B’ (2, 2) C’ (4, 1)

Transformation (x, y) (x - 3, y + 0) Pre-image A (-2, 4) A’ (-5, 4) B’ B C C’ x Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-5, 4) B’ (-6, 2) C’ (-4, 1)

Transformation (x, y) (x + 0, y - 5) Pre-image Image A (-2, 4) B C x A’ Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-2, -1) B’ (-3, -3) C’ (-1, -4) B’ C’

Transformation (x, y) (x + 0, y + 4) Pre-image A (-2, 4) A’ (-2, 8) B’ C’ A B C x Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-2, 8) B’ (-3, 6) C’ (-1, 5)

Transformation (x, y) (x + 3, y - 4) Pre-image Image A (-2, 4) B C A’ x B’ Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (1, 0) B’ (0, -2) C’ (2, -3) C’

Transformation (x, y) (x + 5, y + 2) Pre-image Image A (-2, 4) B’ C’ B C x Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (3, 6) B’ (2, 4) C’ (4, 3)

Transformation (x, y) (x - 4, y - 5) Pre-image A (-2, 4) A’ (-6, -1) B C x A’ Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-6, -1) B’ (-7, -3) C’ (-5, -4) B’ C’

Transformation (x, y) (x - 2, y + 3) Pre-image A (-2, 4) A’ (-4, 7) B’ A C’ B C x Pre-image A (-2, 4) B (-3, 2) C (-1, 1) Image A’ (-4, 7) B’ (-5, 5) C’ (-3, 4)

Reflections You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image. Example: The figure is reflected across line l . l

Reflection Notation Describes the reflection across the x-axis Describes the reflection across the y-axis Describes the reflection across x = 1 Describes the reflection across y = x        

Reflections Reflection across the x-axis: the x coordinates stay the same and the y coordinates change sign. (x , y)  (x, -y) Reflection across the y-axis: the y coordinates stay the same and the x coordinates change sign. (x , y)  (-x, y) Reflection across the y=x: the x and y coordinates stay the same, but you flip each coordinate. (x , y)  (y, x)

Reflecting across vertical lines (x = a) Reflect across x = 2 A B B' A' D C C' D'

Reflect across y = -3 H’(-12, 2) A’(7, -7) T’(2, -7) H T A

Reflecting across y-axis Pre-image Image C'(3, 7) C A T C’ C(-3, 7) A(-3, 2) A'(3, 2) A’ T’ T(2, 2) T'(-2, 2) What do you notice about the x and y coordinates of the pre-image and image points?

Reflecting across x-axis Reflect the following shape across the x-axis Pre-image Image M(2, 1) M’(2, -1) A(-1, 1) A’(-1, -1) T H T(-3, 5) T’(-3, -5) A M H(4, 5) H’(4, -5) A’ M’ T’ H’ What do you notice about the x and y coordinates of the pre-image and image points?

Reflecting across the line y = x Pre-Image Image F(-3, 0) F‘(0, -3) I(4, 0) I’ S’ I'(0, 4) S(4, -9) F I S'(-9, 4) F’ H(-3, -9) H'(-9, -3) H’ What do you notice about the x and y coordinates of the pre-image and image points? H S

Reflect across y = –x M(-5, 2) O(-2, 2) V(0, 6) E(-7, 6) E’ E V M’ M’(-2, 5) O’(-2, 2) V’(-6, 0) E’(-6, -7) M O O’ V’

Lines of Symmetry If a line can be drawn through a figure so the one side of the figure is a reflection of the other side, the line is called a “line of symmetry.” Some figures have 1 or more lines of symmetry. Some have no lines of symmetry. Four lines of symmetry One line of symmetry Two lines of symmetry Infinite lines of symmetry No lines of symmetry

Composite Reflections If an image is reflected over a line and then that image is reflected over a parallel line (called a composite reflection), it results in a translation. Example: C B A Image A reflects to image B, which then reflects to image C. Image C is a translation of image A

Rotations Rotation: a transformation that turns a figure about a fixed point called the center of rotation. An object and its rotation are the same shape and size, but the figures may be turned in different directions. All rotations will be COUNTERCLOCKWISE unless otherwise noted.

Rules for Rotations

Every time we rotate a figure 90 degrees, we switch the order of the coordinates, then switch the sign of the first one.

Example 1:

Example 2:

Example 3:

Dilations A dilation is a transformation which changes the size of a figure but not its shape. This is called a similarity transformation. Since a dilation changes figures proportionately, it has a scale factor k. If the absolute value of k is greater than 1, the dilation is an enlargement. If the absolute value of k is between 0 and 1, the dilation is a reduction. If the absolute value of k is equal to 0, the dilation is congruence transformation. (No size change occurs.)

If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of –k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation.

Example Quadrilateral ABCD has vertices A(-2, -1), B(-2, 1), C(2, 1) and D(1, -1). Find the coordinates of the image for the dilation with a scale factor of 2 and center of dilation at the origin. C’ B’ B C A D A’ D’

Example: F(-3, -3), O(3, 3), R(0, -3) Scale factor 1/3 O O’ F’ R’ F R

Example: T(-1, 0), H(1, 0), I(2, -2), S(-2, -2) Scale factor 4 T H H’

Example 3: Drawing Dilations in the Coordinate Plane Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation with a scale factor of centered at the origin. The dilation of (x, y) is

Example 3: Drawing Dilations in the Coordinate Plane Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation with a scale factor of centered at the origin. The dilation of (x, y) is

Graph the preimage and image. Example 3 Continued Graph the preimage and image. P P’ Q’ R’ R Q