Questions and Answers
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.
Rotate 180° about the origin, then reflect over the x-axis.
These are examples of _________.
Non-rigid transformations
These are examples of _________.
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State the rule for the transformation: first (x,y)→(x,-y), then (x,y)→(x-5,y).
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State the rule for the transformation: first (x,y)→(x+10,y), then (x,y)→(x,-y).
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State the rule for the transformation: first (x,y)→(x+1,y-3), then (x,y)→(-x,y).
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State the rule for the transformation: (x,y)→(-x,-y).
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State the rule for the transformation: first (x,y)→(-x,y), then (x,y)→(x,y-4).
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State the rule for the transformation: first (x,y)→(x,y+4), then (x,y)→(-x,y).
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Describe the composition of transformations: First the triangle was rotated 180° about the origin, then reflected over the y-axis.
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Mirror image over a given line is known as __________.
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Moving a shape about a fixed point (the origin is point (0,0)) is called __________.
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Sliding all the points of a shape according to given directions (up/down changing y, right/left changing x) is known as __________.
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The small hash mark that you put next to the point labels of the image (ie: A') is called __________.
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To reflect a point over the x-axis, the transformation is expressed as __________.
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To reflect a point over the y-axis, the transformation is expressed as __________.
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To reflect a point over the line y=x, the transformation is expressed as __________.
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To reflect a point over the line y=-x, the transformation is expressed as __________.
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To rotate a point about the origin (point (0,0)) 90° Clockwise, the transformation is expressed as __________.
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To rotate a point about the origin (point (0,0)) 180° Clockwise, the transformation is expressed as __________.
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To rotate a point about the origin (point (0,0)) 90° Counter-Clockwise, the transformation is expressed as __________.
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To rotate a point about the origin (point (0,0)) 360° Clockwise, the transformation is expressed as __________.
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If you compose two reflections over parallel lines that are h units apart, it is equivalent to a single translation of __________.
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If you compose two reflections, one over each axis, the final image is a rotation of __________ of the original about the origin.
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If you compose two reflections over lines that intersect at x, the resulting image is a rotation of __________.
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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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Description
Explore the fundamental concepts of geometric transformations, including reflections, translations, and rotations. Test your understanding of composition of transformations by describing each transformation step. Perfect for students learning about transformations in coordinate geometry.
