unit 6: transformations translations
TRANSCRIPT
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Unit 6: Transformations
Translations
Objectives: SWBAT translate point and images on a coordinate plane
A. Identifying Transformations
Translation ~ (AKA a slide) moves all points of an image the same distance in the
same directions
Preimage [A] ~ an image before
any transformations occur.
Image [Aโ]~ the image after any
transformations occurs
Isometry ~ study of congruent transformation
Also known as rigid transformations.
Coordinate Notation ~ (๐, ๐) โ (๐ + ๐, ๐ + ๐) where x and y are the original
Coordinates, and the a, and b are the shifts
Consider the translation that is defined by the coordinate notation (x, y) (x + 2, y โ 3)
1. What is the image of (1, 2)? (๐, ๐) โ (๐ + ๐, ๐ โ ๐) ๐จ๐๐๐๐๐: (๐, โ๐)
2. What is the image of (3, -4)? (๐, โ๐) โ (๐ + ๐, โ๐ โ ๐) ๐จ๐๐๐๐๐: (๐, โ๐)
3. What is the pre-image of (5, 8)? (๐, ๐) โ (๐ โ ๐, ๐ + ๐) ๐จ๐๐๐๐๐: (๐, ๐๐)
4. What is the pre-image of (0, -4)? (๐, โ๐) โ (๐ โ ๐, โ๐ + ๐) ๐จ๐๐๐๐๐: (โ๐, โ๐)
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Draw the following segments after the translation.
6. (x, y)(x + 1, y โ 5) 7. (x, y)(x + 5, y + 3)
Blue is pre-image, move right 1, Blue is pre-image, move right 5,
Down 5 to get to the image. Up 3 to get to the image
8. Graph the image after a translation of โจโ1 , โ4 โฉ 9. Find LMN , where
๐บ๐๐๐ ๐จ๐ (๐, ๐) โ (๐ โ ๐, ๐ โ ๐) with a translation of (3, 1) ,
and then graph the image.
Black is pre-image, move left 1, The given information is the image
Down 4 to get to the image. We need to do the opposite of the
Opposite of the Translation to find
the pre image so subtract 3 and 1.
' 0,4 , ' 2, 1 , ' 2,0L M N
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Reflections Objectives: SWBAT find the reflections of images and points over lines and axis
Reflection~
Flip an image over a line called the Line of Reflection.
Each point of the preimage and its image are
the equidistant from the line of the reflection.
Line of reflection:
A line where an image is reflected over.
Transformations in the Coordinate Plane Rules
Reflection over the ๐ฅ ๐๐ฅ๐๐ : (๐, ๐) โ (๐, โ๐)
Reflection over the ๐ฆ ๐๐ฅ๐๐ : (๐, ๐) โ (โ๐, ๐)
Reflection over the line ๐ฆ = ๐ฅ (๐, ๐) โ (๐, ๐)
1. Draw the image of segment that is a:
Use the rules above to draw the segments. Start at the Red.
a. Reflection in the ๐ฅ โ ๐๐ฅ๐๐
Green Line
b. Reflection in the ๐ฆ โ ๐๐ฅ๐๐
Orange Line
c. Reflection in the line ๐ฆ = ๐ฅ
Magenta Line
d. Reflection in the ๐ฆ โ ๐๐ฅ๐๐ , followed by a reflection in the ๐ฅ โ ๐๐ฅ๐๐
Magenta Line
e. Reflection in the ๐ฅ โ ๐๐ฅ๐๐ , followed by a reflection in the ๐ฆ โ ๐๐ฅ๐๐ .
Magenta Line
f. Reflection in the line ๐ฆ = ๐ฅ, followed by a Reflection in the x-axis.
Orange Line
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A. Name the point. then plot itsโ reflection over the x axis, y axis, and y=x.
Use the rules above to draw the segments.
๐ฅ ๐๐ฅ๐๐ ๐ฆ ๐๐ฅ๐๐ ๐ฆ = ๐ฅ
1. A (3, 2) (๐, โ๐) (โ๐, ๐) (๐, ๐)
2. B (5, 0) (๐, ๐) (โ๐, ๐) (๐, ๐)
3. C (-3, -1) (โ๐, ๐) (๐, โ๐) (โ๐, โ๐)
4. Plot each point, then plot itsโ reflection in the line ๐ฆ = 2. Name the point.
a. M (4, 4) ๐โฒ(๐, ๐)
b. N (-5, 2) ๐โฒ(โ๐, ๐)
c. P (-2, -4) ๐โฒ(โ๐, ๐)
3. Describe the composition of the transformation using coordinate notation for
the above translations .
a. M (4, 4) (๐, ๐) โ (๐, ๐ โ ๐)
b. N (-5, 2) (๐, ๐) โ (๐, ๐)
c. P (-2, -4) (๐, ๐) โ (๐, ๐ + ๐๐)
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Rotations Objectives: SWBAT find and rotate images and points over center of rotation
Rotation~ A โturnโ
is a transformation around a fixed point called the center of rotation, through a
specific angle, and in a specific direction.
Each point of the preimage and its image are the same distance from the center.
Center of Rotation:
The fixed point a rotation turns around
Angle of Rotation:
The amount of turning of a rotation
Clockwise~
same direction as a clock's hands
Counter Clockwise~
Opposite same direction as a clock's hands
Complete the following sentences.
1. A clockwise rotation of 45 around P maps A onto _B_.
2. A clockwise rotation of 90 around P maps C onto _ E_.
3. A clockwise rotation of 180 around P maps _ B __ onto F.
4. A counterclockwise rotation of 90 around P maps H onto _ F _.
5. A counterclockwise rotation of 45 around P maps __F _ onto E.
6. A counterclockwise rotation of 135 around P maps _ B__ onto A.
p
A
B
C
DE
F
G
H
45
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Rotations can be counterclockwise and Clockwise so be careful.
Convert the following clockwise measurements to counterclockwise measurements.
A) 90 degrees is _____270__________ counterclockwise
B) 180 degrees is ____180___________ counterclockwise
C) 270 degrees is ____90___________ counterclockwise
Coordinate Plane and Rotation
Special Rotation Rules
90ยฐ (๐, ๐) โ (โ๐, ๐)
180ยฐ (๐, ๐) โ (โ๐, โ๐)
270ยฐ (๐, ๐) โ (๐, โ๐)
Counter Clockwise ONLY. If you have clockwise you must convert to counter.
Examples
1. Given XYZ with vertices X(2, 0), Y( 7, 0), and Z ( 2, 6). Then rotate it 180
degrees around the origin.
๐ฟ(๐, ๐)
๐( ๐, ๐) ๐ ( ๐, ๐)
๐ฟโฒ(โ๐, ๐)
๐โฒ( โ๐, ๐) ๐โฒ (โ ๐, โ๐)
x axis
y axis
(x,y)
(x',y')
60ยฐ
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Rotate 90ABC counterclockwise around the origin. The Vertices are
A( 0, 2), B ( 3, 1) and C( 4, 3).
๐จ( ๐, ๐)
๐ฉ( ๐, ๐) ๐ช( ๐, ๐)
๐จโฒ( โ๐, ๐)
๐ฉโฒ( โ๐, ๐) ๐ชโฒ(โ๐, ๐)
3. Rotate 270MNO counterclockwise around the origin. The Vertices are
M( 0, 4), N ( 4, 0) and O( 4, 2).
๐ด( ๐, ๐)
๐ต( ๐, ๐) ๐ถ( ๐, ๐)
๐ดโฒ( ๐, ๐)
๐ตโฒ( ๐, โ๐) ๐ถโฒ( ๐, โ๐)
The point ๐(โ2, โ5) is rotated 90ยฐ counter clockwise about the origin, and then the image is
reflected across the line ๐ฅ = 3. What are the coordinates of the final image ๐" ?
Make sure you go in order of the transformations and record each step so you donโt get
lost.
p
Pโ
Rotation
Pโโ
Reflection
๐ท(โ๐, โ๐)
๐ทโฒ(๐, โ๐)
๐ทโฒโฒ( ๐, โ๐)
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x
y
G
K
H
What are the coordinates for the image of โ๐บ๐ป๐พ after a rotation 90ยฐ clockwise about
the origin and a translation of (๐ฅ, ๐ฆ) โ (๐ฅ + 3, ๐ฆ + 2) ?
All the rules are for clockwise, so we need to convert the 90 degree clockwise to
270 degrees counterclockwise.
Pre-Image
Rotation
Translation
๐ฎ(โ๐, โ๐)
๐ฎโฒ(โ๐, ๐)
๐ฎโฒโฒ(๐, ๐)
๐ฏ(๐, โ๐)
๐ฏโฒ(โ๐, โ๐)
๐ฏโฒโฒ(โ๐, ๐)
๐ฒ(๐, โ๐)
๐ฒโฒ(โ๐, โ๐)
๐ฒโฒโฒ(๐, ๐)
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Rotating around a point other than the origin
1. Write down your objects key points (if it is a triangle, put down the 3 vertices)
2. Subtract the point you are rotating around from each point (mind your PEMDAS)
3. Apply correct rotation rule
4. Add the point you are rotating back
5. Plot your points.
Figure ABC is rotated 90ยฐ clockwise about the point (2, 0). What are the coordinates of ๐ดโฒ after
the rotation?
Points
6,3
3, 2
4,5
A
B
C
*
int
6,3
2,0
4,3
Subtract the Rotation Po
A
A
*
int
3,2
2,0
1,2
Subtract the Rotation Po
B
B
*
int
4,5
2,0
2,5
Subtract the Rotation Po
C
C
*
*
*
,
4,3 3, 4
1,2 2, 1
2,5 5, 2
Apply Rotation Rule
y x
A
B
C
int
3,4
2,0
' 5,4
Add Back the Rotation Po
A
A
int
2, 1
2,0
' 4, 1
Add Back the Rotation Po
B
A
int
5, 2
2,0
' 7, 2
Add Back the Rotation Po
C
C
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7. Graph XYZ with vertices X(2, 0), Y( 7, 0), and Z ( 2, 6). Then rotate it 180 degrees around (2, -3).
*
*
*
Points
2,0 2, 3 0,3
7,0 2, 3 5,3
2,6 2, 3 0,9
Rotation
X X
Y Y
Z Z
*
*
*
0,
,
03
5,3
0,9
, 3
5, 3
0, 9
Apply Rotation Rule
x y
X
Y
Z
*
*
*
int
2, 3 ' 2, 6
2
0,
, 3 ' 3, 6
2, 3 ' 2, 12
3
5, 3
0, 9
Add Back the Rotation Po
X X
Y Y
Z Z
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Dilations Objectives: SWBAT reduce or enlarge a figure by a scale factor of k.
Dilation~ enlarges or reduces the original figure proportionally
Multiply each part of the image by the scale factor to get the dilated image.
Scale Factor~ โkโ the amount to enlarge each point
๐ > ๐ โ ๐๐๐๐๐๐๐๐๐๐๐ ๐ < ๐ โ ๐๐๐ ๐๐๐๐๐๐
Examples
A. Graph AB with vertices (0,2)A and (2,1)B . Then find the coordinates of the dilation
image of AB with scale factor of 3, and graph its dilation image.
๐จ(๐, ๐) โ ๐(๐, ๐) โ ๐จโฒ(๐, ๐) ๐ฉ(๐, ๐) โ ๐(๐, ๐) โ ๐ฉโฒ(๐, ๐)
B. Graph DEF with vertices (3,3)D , (0, 3)E and ( 6,3)F . Then find the coordinates
of the dilation image with scale factor of 1
3 and graph its dilation image.
๐ซ(๐, ๐) โ๐
๐(๐, ๐) โ ๐ซโฒ(๐, ๐)
๐ฌ(๐, โ๐) โ๐
๐(๐, โ๐) โ ๐ฌโฒ(๐, โ๐)
๐ญ(โ๐, ๐) โ๐
๐(โ๐, ๐) โ ๐ญโฒ(โ๐, ๐)
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C. Graph image with vertices '( 1,1)A , '( 2,2)B , '(2,2)C and '(1,1)D with ๐ = 0.5. Find
the coordinates of the dilation pre-image and graph its pre-image.
In this example, you get the image, work backwards to find the pre-image.
๐จโฒ(โ๐, ๐) โ ๐(โ๐, ๐) โ ๐จ(โ๐, ๐)
๐ฉโฒ(โ๐, ๐) โ ๐(โ๐, ๐) โ ๐ฉ(โ๐, โ๐) ๐ชโฒ(๐, ๐) โ ๐(๐, ๐) โ ๐ช(๐, ๐) ๐ซโฒ(๐, ๐) โ ๐(๐, ๐) โ ๐ซ(๐, ๐)
Apply the dilation ๐ท: (๐ฅ, ๐ฆ) โ (4๐ฅ, 4๐ฆ) to the triangle given vertices. Which of the following
is the perimeter of the image โ๐ดโฒ๐ตโฒ๐ถโฒ ?
In this example, ๐ซ: (๐, ๐) โ (๐๐, ๐๐) means the same thing as a ๐ = ๐
Perimeters of dilations are also proportional to the scale factor, so you have
two options, find the dilated sides, and add them up. Or find the perimeter
currently, and multiply it by the scale factor.
What is the scale factor for the dilation of ฮ๐ด๐ต๐ถ to image ฮ๐ดโฒ๐ตโฒ๐ถโฒ ?
๐ = ๐
y
x
A'
A
B'
C'
C
B
x
y
C
A B
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Transformation Rules
REFLECTION
Explanation
OVER
๐ = ๐
OVER
๐ ๐๐๐๐
OVER
๐ ๐๐๐๐ LINE
๐ = ๐จ ๐ต๐๐๐๐๐ LINE
๐ = ๐จ ๐ต๐๐๐๐๐
A mirror image over a
line
(๐, ๐) โ (๐, โ๐) (๐, ๐) โ (โ๐, ๐) (๐, ๐) โ (๐, ๐) Must Count to
the line
Must Count to
the line
When canโt I count?
Canโt count with you go over the line ๐ = ๐
ROTATION
Explanation
90 Degrees Counterclockwise
180 Degrees Counterclockwise
270 Degrees Counterclockwise
Spinning around a fixed point
(๐, ๐) โ (โ๐, ๐) (๐, ๐) โ (โ๐, โ๐) (๐, ๐) โ (๐, โ๐)
What do I do when
I donโt rotate around the origin
Subtract the Point of Rotation Do the Rule
Add the Rule Back In
DILATION
Explanation
SHRINK GROW Coordinate
Form
Shrinking or
Enlarging an image
K value is in-between 0 and 1
๐ < ๐ < ๐
K value is greater than 1
๐ > ๐
(๐, ๐) โ (๐๐, ๐๐)
TRANSLATION
Explanation
Coordinate Form Vector Form
A slide
Shifting the image left / right, up / down
(๐, ๐) โ (๐ ยฑ ๐, ๐ ยฑ ๐) (๐, ๐) or โจ๐, ๐โฉ