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Transformations

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Anelise Larcuente

on 10 February 2014

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Transcript of Transformations


Transformations
the act, process, or result of transforming or mapping.
Transformations
There are 4 types of transformations which are:
Ro
t
ations -
t
urn
Re
fl
ections -
fl
ip
Tran
sl
ations -
sl
ide
Di
la
tions -
la
rge,
l
ittle
The first 3 transformations (rotations, reflections, and translations) are
isometries.
Gilde Reflections and compositions are also examples of an isometry. An isometry is a transformation in which the original figure and its image are
congruent
, or equal in size and shape. The original figure is called the
preimage,
and the figure after the transformation is called the
image.
Rotations
a transformation that turns a figure about a fixed point
Rotation rules:
R 90° (x, y) = (-y, x)
R 180° (x, y) = (-x, -y)
R 270° (x, y) = (y, -x)
R -90° (x, y) = (y, -x)
By: Gracie Oney & Anelise Laracuente
Reflections
Translations
Dilations
a transformation in which a geometric figure is reflected across a line, creating a mirror image
a transformation of the plane that slides every point of a figure the same distance in the same direction
a transformation in which a figure is enlarged or reduced using a scale factor, without altering the center point
Reflection rules:
r
x-axis
(x, y) = (x, -y)
r
y-axis
(x, y) = (-x, y)
r
y=x
(x, y) = (y, x)
r
y=-x
(x, y) = (-y, -x)
Translation rules:
T
a,b
(x, y) -> (x + a, y + b)
T
a,b
(x, y) -> <a, b>
Dilation rule: D
k
(x, y) = (kx, ky)
Compositions & Glide Reflection
Glide Reflection- a transformation consisting of a translation combined with a reflection about a parallel line to the direction of the translation
Composition - a transformation that is a combination of a translation, followed by a reflection
Rigid Motion -
the action of taking an object and moving it to a different location without altering its shape or size. Rotations, reflections, translations, glide reflections, and compositions are all examples of rigid motions
A rotation can move 2 ways, counterclockwise and clockwise. If the direction is not specified, then ALWAYS move counterclockwise.
If the image, after being rotated, looks identical to the pre-image, that means that the polygon has
rotational symmetry.
The number of times a polygon has rotational symmetry before 360° is called the
order.
The order pf a shape is the same amount as the number of verticies a polygon has. No image can ever have an order of only 1 since that would mean that the image had rotational symmetry only at 360°. ex - a square's order is 4 (90°, 180°, 270°, 360°), but a pentagon doesn't have rotational symmetry because it is only identical to itself at 360°
Key Words
angle -
the space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet
circle -
a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center)
perpendicular line -
2 lines that are crossed that make a 90 degree angle
parallel line -
2 lines on a plane that never meet and that are always the same distance apart
line segment -
a part of a line that is bounded by two distinct end points
plane -
a flat surface with no thickness that extends on forever
fixed point -
the point that is NOT moved by a given transformation. The fixed point can go from being the origin to just being a random point on the grid.
When rotating a point 90 degrees, just flip the coordinates and multiply y by -1. This rule applies for every 90 degree rotation
about the origin
. If rotating clockwise, flip the coordinates once again, but this time multiply x by -1.
Both the pre-image and image are an equal distance from the line.
Reflecting across different lines:
When a reflection occurs over the line y=x all you have to do is flip your coordinates and nothing more, to get your image. For the line y=-x, you still flip the coordinates, but this time multiply both x and y by -1.
Reflection across y=x:
(4, 8) is the pre-image and (8, 4) is the image.
When reflection across y=0, or the x-axis, all that needs to be done is to multiply y by -1. For the y-axis, or x=0, the opposite is done. This means you multiply x by -1.
However, when reflecting across lines such as x=5, the rule is different. An easy way to do it is do count however many units the point is to the line, and then go the opposite way form the line however many units.
Reflection across x=5:
(4,1) = pre-image. This is one unit away from x=5, so count the other way 1 unit and you get (6,1) for your image!
Rotational Symmetry, Symmetry, & Order
Example -
Rotate point (3,2) 180° about origin
(3, 2) - pre-image
(2, -3) - 90°
(-3, -2) - 180°
Prime and Vector Notation
Vector notation is written as T
a,b
(x, y) -> <a, b>
Prime notation is written as T
a,b
(x, y) -> (x + a, y + b)
The arrows in the translations rules mean "translated." The arrows are basically saying that one point is translated to another.
Example:
Translate point (2, -2) 4 units left and 2 units down
T
4,2
(
2, -2) -> (2 + 4, -2 - 2)
2+4= 6, -2 -2 = -4
new coordinates: 6, -4
In the dilation rule, "K" means the scale factor.
Dilations is the only transformation that is not an isometry.
If the scale factor is anything above 1, then the polygon is being enlarged. If the scale factor is below 1, then it is being reduced.
Examples:
D
3
(1, 3) = (
3
1,
3
3)
coordinates: (3, 9)
D
1/3
(1, 3) = (
1/3
1,
1/3
3)
coordinates: (0.33, 1)
When doing 2 transformations in one problem (like a reflection then another reflection) always do the second transformation on the image of the first transformation.
ex - reflection across y=x then reflection over y=0. The pre-image for y=0 is the image for y=x
A shapes number of
lines of symmetry
depends on how many times the shape can be 'cut' so that both sides are equal. Instead of counting how many lines of symmetry a shape has, all you have to do it count how many sides it has. For a star although, you count the verticies. Even though a circle has 0 sides, the order is infinite.
In vector and prime notations, a and b stand for the number of units a point is being translated. A indicates horizontal (as in 3 units left) and B indicates vertical (as in 7 units down)
With Glide Reflections and Compositions, the same rules that you would do with translations and reflections still apply. Both Glide Reflections and Compositions are isometries. A Glide Reflection is a reflection, followed by a translation. The line of the translation is parallel to the line of reflection. So, if the line of reflection is vertical, then the translation will go up or down as well and vice versa. A Composition is just the opposite. This transformation consists of a translation FOLLOWED by a reflection. The symbol for a composition of transformations is an open circle. The Compositons rule is written as R
x-axis
° T
3,4,
BUT this process is done in reverse. Remember, the translation comes first, then the reflection.
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