Geometric Transformations Math 9

1.) Translations A slide! Moving the shape left or right and/or up or down. The shape can move horizontally (left/right) called a Horizontal Translation (HT). The shape can move vertically (up/down) called a Vertical Translation (VT). And the shape could have both a HT and a VT.

Translations Translations occur by adding or subtracting values from our original points. Mapping rule: (x, y)  (x + HT, y + VT) Ex: Original points A (0, 0), B (1, 1), and C (0, 1) if we have a translation of right 2 (HT of +2) and down 1 (VT of -1). (x, y)  (x + 2, y + - 1) A (0, 0)  (0 + 2, 0 – 1) the new point is (2, -1) A’ B (1, 1)  (1 + 2, 1 – 1 ) the new point is (3, 0) B’ C (0, 1)  (0 + 2, 1 – 1) the new point is (2, 0) C’ Let’s graph it and see what it looks like (on board)

Translations You try one! What will be the new position of the graph with original points A (0, 0), B (1, 1), and C (0, 1) with a translation of left 5 and up 3? How will the shape move? the shape will move backwards 5 (HT of - 5) and up 3 (VT of +3) What will the mapping rule be? (x, y)  (x – 5, y + 3) What will the new points be? A (0, 0)  (0 – 5, 0 + 3)(-5, 3) A’ B (1, 1)  (1 – 5, 1 + 3) (-4, 4) B’ C (0, 1)  (0 – 5, 1 + 3) (-5, 4) C’

2.) Reflections A flip! “Mirror”ing the shape! There can be a reflection over the x-axis (the y-values go negative). There can be a reflection over the y-axis (the x-values go negative).

Reflections Reflections over an axis occur by making one of the values (either x or y) the opposite sign! Mapping rule of a x-axis reflection: (x, y)  (x, - y) the negative indicates the sign of the y values change sign. Mapping rule of a y-axis reflection: (x, y)  (- x, y) the negative indicates the sign of the x values change sign. Note: a common mistake in that a y-axis reflection it is the x-values that change, NOT the y-values.

Reflections Ex: Original points A (0, 0), B (1, 1), and C (0, 1) with a reflection of the x - axis. (x, y)  (x, - y) A (0, 0)  (0, - 0)(0, 0) A’ B (1, 1)  (1, - 1) (1, -1) B’ C (0, 1)  (0, - 1) (0, -1) C’ Let’s graph it and see what it looks like (on board) Ex: Original points D (3, 3), E (4, 5), and F (-2, 1) with a reflection of the y - axis. (x, y)  (- x, y) D (3, 3)  (- 3, 3)(-3, 3) D’ E (4, 5)  (- 4, 5) (-4, 5) E’ F (2, 1)  (- - 2, 1) (2, 1) F’ Let’s graph it and see what it looks like (on board)

Reflections You try one! What will be the new position of the graph with original points G (5, 2), H (-3, 2), and I (0, 1) with a reflection about the y-axis? How will the shape reflect? the shape will reflect across the y-axis. The x values will become opposite signs What will the mapping rule be? (x, y)  (- x, y) What will the new points be? G (5, 2)  (- 5, 2)(-5, 2) G’ H (-3, 2)  (- -3, 2) (3, 2) H’ I (0, 1)  (- 0, 1) (0, 1) I’

Reflections So in a y-axis reflection what changes? The x-values have opposite signs! The shape mirrors itself over the y-axis, flips over to the left or right. In a x-axis reflection what changes? The y-values have opposite signs! The shape mirrors itself over x, flips up or down. Our reflections are all about the opposites… Opposite signs, opposite axes change.

3.) 180°Rotations A turn! Rotations that we will be using rotate around the origin (0, 0). A 180 degree rotation around the origin makes both the x and y values opposite sign! Mapping Rule: (x, y)  (- x, - y) So for all coordinates we just make them the opposite sign!

Rotations Ex: Original points A (0, 0), B (1, 1), and C (0, 1) with a 180° rotation around the origin. (x, y)  (- x, - y) A (0, 0)  (- 0, - 0)(0, 0) A’ B (1, 1)  (- 1, - 1) (-1, -1) B’ C (0, 1)  (- 0, - 1) (0, -1) C’ Let’s graph it and see what it looks like (on board) You try one! Ex: Original points D (3, 3), E (4, 5), and F (-2, 1) with a 180° rotation around the origin. (x, y)  (- x, - y) D (3, 3)  (- 3, - 3)(-3, -3) D’ E (4, 5)  (- 4, - 5) (-4, -5) E’ F (2, 1)  (- - 2, -1) (2, -1) F’ Let’s graph it and see what it looks like (on board)

4.) Dilatations An enlargement or reduction! The entire shape gets bigger or smaller! We just multiply all the original points by the dilatation. We can call the dilatation a stretch factor (S). Mapping Rule: (x, y)  (Sx, Sy) Just multiply!

Dilatations Ex: Original points A (0, 0), B (1, 1), and C (0, 1) with a dilatation of 2. Mapping rule: (x, y)  (2x, 2y) A (0, 0)  (2(0), 2(0))(0, 0) A’ B (1, 1)  (2(1), 2(1)) (2, 2) B’ C (0, 1)  (2(0), 2(1)) (0, 2) C’ Let’s graph it and see what it looks like (on board) Ex: Original points D (2, 3), E (4, 5), and F (0, 4) with a dilatation of ½. Mapping rule: (x, y)  (½x, ½y) D (2, 3)  (½(2), ½(3))(1, 3/2 or 1.5) D’ E (4, 5)  (½(4), ½(5)) (2, 5/2 or 2.5) E’ F (0, 4)  (½(0), ½(4)) (0, 2) F’

Review of the Transformations 1.) Translations A slide (x, y)  (x + HT, y + VT) 2.) Reflections A flip x-axis reflection: (x, y)  (x, - y) y-axis reflection: (x, y)  (- x, y) 3.) 180° Rotations A turn both are opposite: (x, y)  (- x, - y) 4.) Dilatations An enlargement: (x, y)  (Sx, Sy) where S is more than 1. A reduction:(x, y)  (Sx, Sy) where S is between 0 and 1.