transformations & coordinate geometry

Transformations & Coordinate Geometry

Transformations & Coordinate GeometryYou Should Learn:

Some basic properties of transformations and symmetry

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

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

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

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’

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’

Transformations – Model Motion

Translation – Glide or SlideRotation – (about an axis)Reflection – Mirror imageDilation – larger or smaller

Rigid Transformations

Translation

Rotation

Reflection

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

Translation

Pre-ImageImage Slide Arrow

A

B CA’

B’ C’

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 )

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)

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

Rigid Transformations

Translation

Rotation

ReflectionPre-Image ImageHorizontal Translation

( x , y )

Vertical Translation ( x , y )

( x + a, y )

( x , y + b )

Rigid Transformations

Reflections

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.

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)

Reflection

Mirror Line

Pre-Image

Image

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’

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

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

Rigid Transformations

Rotations

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.

Rotations

clockwise

counterclockwise

Keep in mindRotation are counterclockwise unless otherwise stated

Rotation – 90° 180° 270° 45° ? °

Pre-Image

Image90°

Image180°

Image

270°

Note: This Example Rotation is Clockwise

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

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 )

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

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

(+,+)

(+,-)(- , -)

(- , +)

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