4.4 reflections and rotations.ppt
TRANSCRIPT
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4.4 Transformations with
Matrices
2. Reflections and Rotations
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2) Reflections
A reflection, or flip, is a transformation that
creates symmetry.
You can use matrix multiplication to graph
reflections in the coordinate plane.
There are four reflection matrices you are
responsible for knowing.
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2) Reflections
Reflection in the y-axis Reflection in the x-axis
−
1
1
−1
1
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2) Reflections
Reflection in the line y ! x Reflection in the line y ! " x
1
1
−
−
1
1
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Example 1# $i%en triangle ABC with A &'(, 1),
B &' 2, *) and C &, 2), reflect the triangle
across the y"axis. Then, sketch the image.
2) Reflections
A B C
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Example 1# $i%en triangle ABC with A &'(, 1),
B &' 2, *) and C &, 2), reflect the triangle
across the y"axis. Then, sketch the image.
2) Reflections
y- axis reflection
matrix
A B C
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Example 1# $i%en triangle ABC with A &'(, 1),
B &' 2, *) and C &, 2), reflect the triangle
across the y"axis. Then, sketch the image.
2) Reflections
y- axis reflection
matrix
A B C A’ B’ C’
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2) Reflections
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2) Reflections
Example 2: $i%en triangle ABC where A &'(, 1), B &' 2, *)
and C &, 2), reflect the triangle across the x"axis. Then,
sketch the image.
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2) Reflections
Example 2: $i%en triangle ABC where A &'(, 1), B &' 2, *)
and C &, 2), reflect the triangle across the x"axis. Then,
sketch the image.
A B C
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2) Reflections
Example 2: $i%en triangle ABC where A &'(, 1), B &' 2, *)
and C &, 2), reflect the triangle across the x"axis. Then,
sketch the image.
x- axis reflection
matrix
A B C
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2) Reflections
Example 2: $i%en triangle ABC where A &'(, 1), B &' 2, *)
and C &, 2), reflect the triangle across the x"axis. Then,
sketch the image.
−−−
−−=
−−•
− 2*1
2(
2*1
2(
1
1
x- axis reflection
matrix
A B C A’ B’ C’
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2) Reflections
−−−
−−=
−−•
− 2*1
2(
2*1
2(
1
1
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A rotation is a transformation that turns a figure
about a fixed point called a center of
rotation.
You can rotate a figure as much as +o.
-n this text, all rotations are counterclockwise
about the origin.
2) Rotations
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2) Rotations
Rotation of o Rotation of +o
Rotation of 1/o Rotation of 20o
−
1
1
−−
1
1
− 1
1
1
1
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Example 1# $i%en triangle ABC where A &'(,
1), B &' 2, *) and C &, 2), rotate the triangle
20. Then, sketch the image.
2) Rotations
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Example 1# $i%en triangle ABC where A &'(,
1), B &' 2, *) and C &, 2), rotate the triangle
20. Then, sketch the image.
2) Rotations
A B C
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Example 1# $i%en triangle ABC where A &'(,
1), B &' 2, *) and C &, 2), rotate the triangle
20. Then, sketch the image.
2) Rotations
270o rotation
matrtix
A B C
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Example 1# $i%en triangle ABC where A &'(,
1), B &' 2, *) and C &, 2), rotate the triangle
20. Then, sketch the image.
2) Rotations
270o rotation
matrtix
A B C A’ B’ C’
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2) Rotations
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2) Rotations
Example 2# The matrix below represents the %ertices of
a polygon. rite a matrix to represent the %ertices after a
rotation of o.
A B C D
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2) Rotations
Example 2# The matrix below represents the %ertices of
a polygon. rite a matrix to represent the %ertices after a
rotation of o.
90o rotation
matrtix
A B C D
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2) Rotations
Example 2# The matrix below represents the %ertices of
a polygon. rite a matrix to represent the %ertices after a
rotation of o.
90o rotation
matrtix
A B C D A’ B’ C’ D’
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Homework
1) 3reate some way to remember the 8 matrices
used for reflections and rotations.
You are responsible for knowin all 8.
The matrices are located on p.1!" and p.1!4
2) p.1 41, 11, 1+, 1(, 1/"21, +1, +2
+) #$%& 'E(E*(+, - section 4.4