Transformations & Coordinate Geometry
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Transcript of Transformations & Coordinate Geometry
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Transformations & Coordinate Geometry
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Transformations & Coordinate GeometryYou Should Learn:
Some basic properties of transformations and symmetry
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A rule for moving every point in a plane figure to a new location.
A transformation transforms a geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Transformations
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A rule for moving every point in a plane figure to a new location.
A transformation transforms a geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Transformations
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Terminology
Image – final image after transformationLabeled with “Prime” (Example: A’)Pre-image – image before transformationLabeled with Capital Letters
A A’
B B’
Pre-Image Image
Horizontal Translation
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Terminology
If the image is congruent to the original figure, the process is called rigid transformation, or isometry
A
B
Pre-Image
Horizontal Translation C
A’
B’
Image C’
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Terminology
A transformation that does not preserve the size and shape is called nonrigid transformation
A
B
Pre-Image
Horizontal Translation C
A’
B’
Image C’
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Transformations – Model Motion
Translation – Glide or SlideRotation – (about an axis)Reflection – Mirror imageDilation – larger or smaller
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Rigid Transformations
Translation
Rotation
Reflection
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Rigid Transformations
TraslationsA transformation that moves each point in a figure the same distance in the same directionIn 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
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Translation
Pre-ImageImage Slide Arrow
A
B CA’
B’ C’
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Rigid Transformations
TraslationsTo find any image of any point
Pre-Image Image
Horizontal Translation ( x , y )
Vertical Translation ( x , y )
( x + a, y )( x , y + b )
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A (-2,4)
B (1,6)
C (2,1)
A’ (3,7)
B’ (6,9)
C’ (7,4)
A’ = (-2+5,4+3)
B’ = (1+5, 6+3)
C’ = (2+5, 1+3)
TraslationsRigid Transformations -
(-2,4)
(1,6)
(2,1)
(4,4)
(6,9)
(3,7)
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A rule for moving every point in a plane figure to a new location.
A transformation transforms a geometric figure, shifting it around, flipping it over, rotating it, stretching it, or deforming it.
Transformations
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Rigid Transformations
Translation
Rotation
ReflectionPre-Image ImageHorizontal Translation
( x , y )
Vertical Translation ( x , y )
( x + a, y )
( x , y + b )
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Rigid Transformations
Reflections
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Rigid Transformations
Reflections
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.
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Rigid Transformations
ReflectionsIn graphing, a reflection across the x -axis changes the sign of the y coordinate.
In graphing, a reflection across the y-axis changes the sign of the x coordinate.
(x, y) → (x, -y)
(x, y) → (-x, y)
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Reflection
Mirror Line
Pre-Image
Image
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Rigid Transformations
Reflection
L (-7,5)
M (0,5)
N (-2,1)
O (-5,1)
L’ (-7,-5)
M’ (0,-5)
N’ (-2,-1)
O’ (-5,-1)
LMNO is reflected over the x-axis
L M
NO
-1-2-3-4-5-6-7-1
-2
-3
-4
-5
-6
1
2
3
4
5
N’O’
L’
M’
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Rigid Transformations
Reflection
P (-8,-3)
Q (-2,-3)
S (-2,-6)
R (-8,-6)
P’ (8,-3)
Q’ (2,-3)
S’ (2,-6)
R’ (8,-6)
P Q
SR
-1-2-3-4-5-6-7-1
-2
-3
-4
-5
-6
62 3 4 51 7 8-8
P’Q’
S’ R’-7
PQSR is reflected over the y-axis
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Rigid Transformations
Reflection
P (-8,-3)
Q (-2,-3)
S (-2,-6)
R (-8,-6)
P’ (8,-3)
Q’ (2,-3)
S’ (2,-6)
R’ (8,-6)
P Q
SR
-1-2-3-4-5-6-7-1
-2
-3
-4
-5
-6
62 3 4 51 7 8-8
P’Q’
S’ R’-7
PQSR is reflected over the y-axis
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Rigid Transformations
Rotations
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Rigid Transformations
RotationsIt is performing by "spinning“ the object around a fixed point known as the center of rotation (such as the origin).No change in shape, but the orientation and location change.The distance from the center to any point on the shape stays the same.
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Rotations
clockwise
counterclockwise
Keep in mindRotation are counterclockwise unless otherwise stated
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Rotation – 90° 180° 270° 45° ? °
Pre-Image
Image90°
Image180°
Image
270°
Note: This Example Rotation is Clockwise
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The Rules for rotating a figure about the origin couterclockwise
( x , y )
( x , y )
( x , y )
( x , y )
Þ (- y , x )
Þ ( -x , -y )
Þ ( y , -x )
( nx, ny )
Pre-Image
Image
900 Rotation about Originmultiply the y-coordinate by -1 and then interchange the y- and y-coordinate
1800 Rotation about Originmultiply the x- and y-coordinate by -1
2700 Rotation about Originmultiply the x-coordinate by -1 and then interchange the x- and y-coordinate
ilation
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Rigid Transformations
Rotations
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)
Rotation 1800
about the origin
( x , y )( -x , -y )
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Rigid TransformationsRotation
A (2,5)
B (6,4)
C (6,2)
D (2,2)
A’ (5,-2)
B’ (4,-6)
C’ (2,-6)
D’ (2,-2)
Rotate quadrilateral ABCD 900 clockwise about the origin
A (2,5)
B (6,4)
C (6,2)D (2,2)
A’(5,-2
D’ (2,-2)
C’ (2,-6) B’ (4,-6)
-1-2-3-1
-2
-3
-4
-5
-6
1
2
3
4
5
1 2 3 4 5 6 7
Switch the x, y values of each ordered pair for the location of the new point.
Then, multiply the new y-coordinate by -1
( x , y )( y, -x ) because 900 clockwise = 2700 counterclockwise
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Rigid Transformations
Rotation
-1-2-3-4-5-6-7-1
-2
-3
-4
-5
-6
1
2
3
4
5
1 2 3 4 5 6 7
(+,+)
(+,-)(- , -)
(- , +)
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Graphing Motion
( x , y )
( x , y )
( x , y )
( x , y )
( x , y )
( x , y )
( x , y )
( x , y )
Þ ( x + a, y )
Þ ( x , y + b )
Þ ( x , -y )
Þ ( -x , y )
Þ ( -y , x )
Þ ( -x , -y )
Þ ( y , -x )
( nx, ny )
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