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Pre Algebra
Unit 2: Transformations
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Terms in this set (43)
Function
a relationship between 2 sets of numbers where each member of the first set corresponds to only one member of the second set; a set of ordered pairs where each x-value/element of the domain corresponds with one y-value/element of the range
Mapping
a correspondence between the pre-image and image where the number of of image points is equal to or less than the number of pre-image points
One to One Function
a function where there is the same number of elements in the domain as there is in the range
Transformation
a mapping in which there is the same number of points in the pre-image as there is in the image; a change in the position, size, or shape of a figure
How Mappings are Related to Functions
-mapping (geometry) = function (algebra)
-pre-image (geometry) = domain/input (algebra)
-image (geometry) = range/output (algebra)
-The number of pre-image points should be equal to or greater than the number of image points (geometry) = The number of domain values should be equal to or greater than the number of range values (algebra)
-transformation (geometry) = one to one function (algebra)
Isometric Transformation/Rigid Motion
a transformation that preserves the lengths of segments and the measures of angles of a figure from pre-image to image
Transformations that are Isometric
Rotations, Reflections, Translations
Transformations that are Not Isometric
Dilations and Stretches
Symmetry
the quality a figure has of being able to carry itself onto itself
Types of Symmetry
line/reflection symmetry, rotational symmetry, point symmetry
Line/Reflection Symmetry
type of symmetry where the figure can be mapped onto itself by a reflection in a line
Number of Lines of Symmetry in Regular Polygons
number of lines of symmetry = number of sides
-Odd number of sides: a line of symmetry through each vertex
-Even number of sides: a line of symmetry through each vertex and midpoint of each side
Rotational Symmetry
type of symmetry where the figure is the image of itself under a rotation about a point through any angle whose measure is between 0 and 360 degrees (0 and 360 degrees are excluded from counting as having rotational symmetry because it represents the starting position)
Angle of Rotational Symmetry
smallest angle through which a figure can be rotated so it will coincide with itself (will always be a factor 360)
Order
the number of times the figure coincides with itself in 360 degrees
Rules about Order
-A figure must have an order of at least 2 to have rotational symmetry
-The order of rotational symmetry of any regular polygon is the same as the number of lines of symmetry it has and the number of sides it has.
-(Angle of rotational symmetry)(order) = 360 degrees
Point Symmetry
type of symmetry where a figure is built around a point such that every point in the figure has a matching point that is the same distance from the central point but in the opposite direction
How to Test for Point Symmetry
Turn a figure 180 degrees (upside-down) and see if it looks the same. If it does, then it has point symmetry.
Figures that have rotational symmetry with an order of 2 have point symmetry
Rotation
an isometric transformation that turns a figure about a fixed point
Center of Rotation
the fixed point about which a figure rotates, usually the origin
Angle of Rotation
rays drawn from the center of rotation to a point and its image
-A positive angle of rotation turns the figure counterclockwise
-A negative angle of rotation turns the figure clockwise
Rotation Notation
R (center of rotation), angle of rotation (Pre-image)
*The origin is usually the center of rotation, written as (0,0) or o
*Center of rotation and angle of rotation are written in subscripts
Rotation Rules
90 degrees counterclockwise/270 degrees clockwise: (-y,x)
180 degrees: (-x,-y)
270 degrees counterclockwise/90 degrees clockwise: (y,-x)
Properties Preserved by Rotations
1. Distance (lengths of segments stay the same)
2. Angle measures
3. Parallelism (parallel lines remain parallel)
4. Collinearity (points stay on the same lines)
5. Orientation (lettering order stays the same)
Construction: Rotation
1. Draw a circle from center of rotation to a pre-image point
2. Repeat for all pre-image points
3. Draw a segment from each point to the center
4. Use protractor to measure the rotational degree
5. Draw image point on the circle
Construction: Center of Rotation
1. Draw an auxiliary segment between a pre-image point with its corresponding image point
2. Find perpendicular bisector of that segment
3. Repeat steps 1-2 for another pre-image point
4. The point of intersection of the perpendicular bisectors is the center of rotation
Reflection
an isometric transformation in which each point of the pre-image has an image that is the same distance from the line of reflection as the pre-image point but on the opposite side of the line
Reflection Notation
r m (pre-image) --> (image)
*m represents the line of reflection and is written in subscript
Reflection Rules
Reflect across the x-axis: (x,-y)
Reflect across the y-axis: (-x,y)
Reflect across the line y=x: (y,x)
Reflect across the line y=-x: (-y,-x)
Reflect over a vertical or horizontal line: Graph the line, count the distance from the pre-image to the line, then count the same distance but in the opposite direction and graph the image point
Point Reflection through Origin
Notation: r (0,0) (x,y) --> (-x,-y)
Same as a rotation of 180 degrees
Properties Preserved by Reflections
1. Distance (segment lengths)
2. Angle measures
3. Parallelism (parallel lines remain parallel)
4. Collinearity (points stay on the same line)
5. Reflection does not preserve orientation
*Points on the line of reflection are invariant under the reflection
Construction: Reflection
1. Given a line of reflection
2. For each pre-image point, construct a perpendicular bisector through a point off the line from the point to the line of reflection. The point of intersection between the second pair of arcs is the location of the image point
3. Connect the points to form the image figure
Construction: Line of Reflection
1. Draw an auxiliary line connecting a pre-image point to its corresponding image point
2. Do perpendicular bisector construction
*Line of reflection passes through any invariant points
Translation
an isometric transformation that slides an object a fixed distance in a given direction
Vector
used to describe the fixed distance and the given direction with vector notation
Vector Notation
<x,y>
The x-value describes the effect on the x-coordinates
The y-value describes the effect on the y-coordinates
Ways that directions can be given for translations
-Told: "move right 2 units and down 4 units"
-Rule: "(x,y) --> (x+2, y-4)
-Notation: "T 2,-4" (2 and -4 are written as subscripts)
-Vector: "Translate along the vector <2, -4>" or "T<2, -4>" (2 and -4 are written as subscripts)
Construction: Translation
1. Given pre-image figure and one image point
2. Measure distance from a pre-image point to image point with compass
3. Put point of compass on another pre-image point and copy the length (draw an arc)
4. Repeat step 3 for all pre-image points
5. Use the compass the measure the distance between the pre-image point whose image is given and another pre-image point
6. Put point of compass on the given image point and copy the length (draw an arc)
7. Repeat steps 5-6 for all other pre-image points
8. Points of intersection between arcs are where the image points are
Composition of Transformations
a sequence of 2 or more transformations combined
Notation of a Composition of Transformations
r x-axis ○ T <3,4>
-Transformations are performed from right to left (do the rightmost transformation and work to the left)
-You MUST perform the transformations in the order indicated by the notation
Glide Reflection
a commutative composition of a translation and a reflection where the line of reflection is parallel to the direction of the translation so it does not matter whether you translate or reflect first
Double Reflection Over Parallel Lines
a translation
-The distance that the figure moved is twice the distance between the lines
-The direction that the figure moves in is determined by the order of the reflections
-The location of the pre‐image has nothing to do with the direction
Double Reflection Over Intersecting Lines
a rotation
-The angle of rotation is double the acute angle formed between the intersecting lines
-The direction the figure moves in (counterclockwise or clockwise) will be determined by the direction/order of the reflections
-The location of the pre‐image has nothing to do with the direction of the rotation
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