Transformations & Coordinate Geometry
Transformations & Coordinate Geometry You Should Learn: Some basic properties of transformations and symmetry
Transformations A rule for moving every point in a plane figure to a new location. A transformationtransformsa geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Transformations A rule for moving every point in a plane figure to a new location. A transformationtransformsa geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Terminology • Image – final image after transformation Labeled with “Prime” (Example: A’) • Pre-image – image before transformation Labeled with Capital Letters B B’ Horizontal Translation A A’ Pre-Image Image
Terminology • If the image is congruent tothe original figure, the process is called rigid transformation, orisometry B B’ A Horizontal Translation A’ C Pre-Image Image C’
Terminology • A transformation that does not preserve the size and shape is called nonrigid transformation B B’ A Horizontal Translation A’ C C’ Pre-Image Image
Transformations – Model Motion • Translation – Glide or Slide • Rotation – (about an axis) • Reflection – Mirror image • Dilation – larger or smaller
Rigid Transformations Rotation Translation Reflection
Rigid TransformationsTraslations A transformation that moves each point in a figure the same distance in the same direction In a translation a figure slides up or down, or left or right. In graphing translation, all x and y coordinates of a translated figure change by adding or subtracting
Translation A Pre-Image Image Slide Arrow C B A’ B’ C’
Rigid TransformationsTraslations To find any image of any point Pre-Image Image Horizontal Translation ( x , y ) Vertical Translation ( x , y ) ( x + a, y ) ( x , y + b )
Rigid Transformations - Traslations (6,9) (3,7) (1,6) (4,4) (-2,4) (2,1) A (-2,4) B (1,6) C (2,1) A’ = (-2+5,4+3) B’ = (1+5, 6+3) C’ = (2+5, 1+3) A’ (3,7) B’ (6,9) C’ (7,4)
Transformations A rule for moving every point in a plane figure to a new location. A transformationtransformsa geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Rigid Transformations Rotation Translation Image Reflection Pre-Image Horizontal Translation ( x , y ) Vertical Translation ( x , y ) ( x + a, y ) ( x , y + b )
Rigid TransformationsReflections
Rigid TransformationsReflections A transformation where a figure is flipped across a line such as the x-axis or the y-axis. In a reflection, a mirror image of the figure is formed across a line called a line of symmetry. No change in size. The orientation of the shape changes.
Rigid TransformationsReflections In graphing, areflection across the x -axis changes the sign of the y coordinate. In graphing, areflection across the y-axis changes the sign of the x coordinate. • (x, y) → (x, -y) • (x, y) → (-x, y)
Reflection Pre-Image Image Mirror Line
Rigid TransformationsReflection L M 5 4 3 2 O N 1 LMNO is reflected over the x-axis -7 -6 -5 -4 -3 -2 -1 -1 O’ N’ -2 -3 -4 -5 L’ M’ -6 L (-7,5) M (0,5) N (-2,1) O (-5,1) L’ (-7,-5) M’ (0,-5) N’ (-2,-1) O’ (-5,-1)
Rigid TransformationsReflection P Q 1 -8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 -1 -2 R S PQSR is reflected over the y-axis -3 -4 -5 Q’ P’ -6 -7 S’ R’ P (-8,-3) Q (-2,-3) S (-2,-6) R (-8,-6) P’ (8,-3) Q’ (2,-3) S’ (2,-6) R’ (8,-6)
Rigid TransformationsReflection P Q 1 -8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 -1 -2 R S PQSR is reflected over the y-axis -3 -4 -5 Q’ P’ -6 -7 S’ R’ P (-8,-3) Q (-2,-3) S (-2,-6) R (-8,-6) P’ (8,-3) Q’ (2,-3) S’ (2,-6) R’ (8,-6)
Rigid TransformationsRotations
Rigid TransformationsRotations • Itisperformingby "spinning“ the object around a fixed point known as the center of rotation (such as theorigin). • No change in shape, buttheorientation and locationchange. • The distance from the center to any point on the shape stays the same.
Rotations counterclockwise clockwise Keep in mind Rotation are counterclockwise unless otherwise stated
Rotation – 90° 180° 270° 45° ? ° Pre-Image Image 270° Image 90° Image 180° Note: This Example Rotation is Clockwise
( x , y ) ( x , y ) ( x , y ) ( x , y ) (- y , x ) ( -x , -y ) ( y , -x ) ( nx, ny ) The Rules forrotating a figure abouttheorigincouterclockwise Pre-Image Image 900 Rotation about Origin multiply the y-coordinate by -1 and then interchange the y- and y-coordinate 1800 Rotation about Origin multiply the x- and y-coordinate by -1 2700 Rotation about Origin multiply the x-coordinate by -1 and then interchange the x- and y-coordinate ilation
Rigid TransformationsRotations Rotation 1800 abouttheorigin ( x , y )( -x , -y ) A’ (0,-4) B’ (-7,-4) C’ (-9,-2) D’ (-7,0) E’ (0,0) A (0,4) B (7,4) C (9,2) D (7,0) E (0,0)
Rigid TransformationsRotation A (2,5) B (6,4) Rotate quadrilateral ABCD 900 clockwise about the origin ( x , y )( y, -x ) because 900 clockwise = 2700 counterclockwise 5 C (6,2) D (2,2) 4 • Switch the x, y values of each ordered pair for the location of the new point. • Then, multiply the new • y-coordinate by -1 D’ (2,-2) A’(5,-2 3 2 1 -3 -2 -1 1 2 3 4 5 6 7 -1 B’ (4,-6) C’ (2,-6) -2 A (2,5) B (6,4) C (6,2) D (2,2) A’ (5,-2) B’ (4,-6) C’ (2,-6) D’ (2,-2) -3 -4 -5 -6
Rigid TransformationsRotation 5 (+,+) (- , +) 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -1 -2 (+,-) -3 (- , -) -4 -5 -6
( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x + a, y ) ( x , y + b ) ( x , -y ) ( -x , y ) ( -y , x ) ( -x , -y ) ( y , -x ) ( nx, ny ) Graphing Motion Pre-Image Image Horizontal Translation Vertical Translation Reflection through x-axis Reflection through y-axis 900 Rotation about Origin 1800 Rotation about Origin 2700 Rotation about Origin ilation after multiply the y-coordinate by -1 and then interchange the y- and y-coordinate after multiply the x- and y-coordinate by -1 after multiply the x-coordinate by -1 and then interchange the x- and y-coordinate
