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Describe the composition of transformation: First, the triangle was reflected over the x-axis, then translated horizontally 6 units to the right and vertically 2 units up.
Describe the composition of transformation: First, the triangle was reflected over the x-axis, then translated horizontally 6 units to the right and vertically 2 units up.
First reflect over the x-axis, then translate right 6 units and up 2 units.
Describe the composition of transformations: First the triangle was rotated 180° about the origin, then reflected over the x-axis.
Describe the composition of transformations: First the triangle was rotated 180° about the origin, then reflected over the x-axis.
Rotate 180° about the origin, then reflect over the x-axis.
These are examples of _________.
These are examples of _________.
Non-rigid transformations
These are examples of _________.
These are examples of _________.
State the rule for the transformation: first (x,y)→(x,-y), then (x,y)→(x-5,y).
State the rule for the transformation: first (x,y)→(x,-y), then (x,y)→(x-5,y).
State the rule for the transformation: first (x,y)→(x+10,y), then (x,y)→(x,-y).
State the rule for the transformation: first (x,y)→(x+10,y), then (x,y)→(x,-y).
State the rule for the transformation: first (x,y)→(x+1,y-3), then (x,y)→(-x,y).
State the rule for the transformation: first (x,y)→(x+1,y-3), then (x,y)→(-x,y).
State the rule for the transformation: (x,y)→(-x,-y).
State the rule for the transformation: (x,y)→(-x,-y).
State the rule for the transformation: first (x,y)→(-x,y), then (x,y)→(x,y-4).
State the rule for the transformation: first (x,y)→(-x,y), then (x,y)→(x,y-4).
State the rule for the transformation: first (x,y)→(x,y+4), then (x,y)→(-x,y).
State the rule for the transformation: first (x,y)→(x,y+4), then (x,y)→(-x,y).
Describe the composition of transformations: First the triangle was rotated 180° about the origin, then reflected over the y-axis.
Describe the composition of transformations: First the triangle was rotated 180° about the origin, then reflected over the y-axis.
Mirror image over a given line is known as __________.
Mirror image over a given line is known as __________.
Moving a shape about a fixed point (the origin is point (0,0)) is called __________.
Moving a shape about a fixed point (the origin is point (0,0)) is called __________.
Sliding all the points of a shape according to given directions (up/down changing y, right/left changing x) is known as __________.
Sliding all the points of a shape according to given directions (up/down changing y, right/left changing x) is known as __________.
The small hash mark that you put next to the point labels of the image (ie: A') is called __________.
The small hash mark that you put next to the point labels of the image (ie: A') is called __________.
To reflect a point over the x-axis, the transformation is expressed as __________.
To reflect a point over the x-axis, the transformation is expressed as __________.
To reflect a point over the y-axis, the transformation is expressed as __________.
To reflect a point over the y-axis, the transformation is expressed as __________.
To reflect a point over the line y=x, the transformation is expressed as __________.
To reflect a point over the line y=x, the transformation is expressed as __________.
To reflect a point over the line y=-x, the transformation is expressed as __________.
To reflect a point over the line y=-x, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 90° Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 90° Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 180° Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 180° Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 90° Counter-Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 90° Counter-Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 360° Clockwise, the transformation is expressed as __________.
To rotate a point about the origin (point (0,0)) 360° Clockwise, the transformation is expressed as __________.
If you compose two reflections over parallel lines that are h units apart, it is equivalent to a single translation of __________.
If you compose two reflections over parallel lines that are h units apart, it is equivalent to a single translation of __________.
If you compose two reflections, one over each axis, the final image is a rotation of __________ of the original about the origin.
If you compose two reflections, one over each axis, the final image is a rotation of __________ of the original about the origin.
If you compose two reflections over lines that intersect at x, the resulting image is a rotation of __________.
If you compose two reflections over lines that intersect at x, the resulting image is a rotation of __________.
A composition of a reflection and a translation is known as __________.
A composition of a reflection and a translation is known as __________.
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Study Notes
Compositions of Transformations
- Compositions involve performing multiple transformations in sequence on a geometric figure.
- Example: A triangle can first be reflected over the x-axis, then translated right 6 units and up 2 units.
Basic Transformations
- Reflection: Creates a mirror image across a specified line.
- Rotation: Moves a shape around a fixed point (often the origin).
- Translation: Shifts all points of a shape according to specified directions (changes in x and y coordinates).
Rigid vs. Non-Rigid Transformations
- Rigid Transformations: Preserve size and shape (e.g., rotations, reflections).
- Non-Rigid Transformations: Do not preserve size and shape (e.g., dilations).
Transformation Rules
- Reflect over the x-axis: (x,y) → (x, -y)
- Translate horizontally and vertically: applied sequentially, e.g., first (x,y) → (x+10,y) then apply another transformation.
- Composite transformation example: First (x,y) → (x+1,y-3) then apply reflection as (-x,y) resulting in the final position.
Specific Transformation Rules
- To reflect over the y-axis: (x,y) → (-x, y)
- To reflect over the line y=x: (x,y) → (y,x)
- To reflect over the line y=-x: (x,y) → (-y,-x)
- To rotate 90° clockwise about the origin: (x,y) → (y, -x)
- To rotate 180° clockwise about the origin: (x,y) → (-x,-y)
- To rotate 90° counter-clockwise about the origin: (x,y) → (-y,x)
Theorems Related to Reflections
- Reflection over Parallel Lines Theorem: Composing two reflections over parallel lines, h units apart, equals a translation of 2h units.
- Reflection over the Axes Theorem: Two reflections over both axes result in a 180° rotation about the origin.
- Reflection over Intersecting Lines Theorem: Two reflections over intersecting lines yield a rotation of 2x, centered at the point of intersection.
Glide Reflection
- A glide reflection consists of a reflection followed by a translation, where the translation aligns with the direction of the reflection line.
Notation
- Primes (e.g., A') denote the image of a point after a transformation.
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