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 [A’]~ the image after any transformations occurs Isometry ~ study of congruent transformation Also known as rigid transformations. Coordinate Notation ~ ( , ) → ( + , + ) 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 – 3) 1. What is the image of (1, 2)? (, ) → ( + , − ) : (, −) 2. What is the image of (3, -4)? (, −) → ( + , − − ) : (, −) 3. What is the pre-image of (5, 8)? (, ) → ( − , + ) : (, ) 4. What is the pre-image of (0, -4)? (, −) → ( − , − + ) : (−, −)
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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 [A’]~ the image after any
transformations occurs
Isometry ~ study of congruent transformation
Also known as rigid transformations.
Coordinate Notation ~ (𝒙, 𝒚) → (𝒙 + 𝒂, 𝒚 + 𝒃) 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 – 3)
1. What is the image of (1, 2)? (𝟏, 𝟐) → (𝟏 + 𝟐, 𝟐 − 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟑, −𝟏)
2. What is the image of (3, -4)? (𝟑, −𝟒) → (𝟑 + 𝟐, −𝟒 − 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟓, −𝟕)
3. What is the pre-image of (5, 8)? (𝟓, 𝟖) → (𝟓 − 𝟐, 𝟖 + 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟑, 𝟏𝟏)
4. What is the pre-image of (0, -4)? (𝟎, −𝟒) → (𝟎 − 𝟐, −𝟒 + 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (−𝟐, −𝟕)
Draw the following segments after the translation.
6. (x, y)(x + 1, y – 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 ⟨−1 , −4 ⟩ 9. Find LMN , where
𝑺𝒂𝒎𝒆 𝑨𝒔 (𝒙, 𝒚) → (𝒙 − 𝟏, 𝒚 − 𝟒) 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
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 𝑥 𝑎𝑥𝑖𝑠: (𝒙, 𝒚) → (𝒙, −𝒚)
Reflection over the 𝑦 𝑎𝑥𝑖𝑠: (𝒙, 𝒚) → (−𝒙, 𝒚)
Reflection over the line 𝑦 = 𝑥 (𝒙, 𝒚) → (𝒚, 𝒙)
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 𝑥 − 𝑎𝑥𝑖𝑠
Green Line
b. Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠
Orange Line
c. Reflection in the line 𝑦 = 𝑥
Magenta Line
d. Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠, followed by a reflection in the 𝑥 − 𝑎𝑥𝑖𝑠
Magenta Line
e. Reflection in the 𝑥 − 𝑎𝑥𝑖𝑠, followed by a reflection in the 𝑦 − 𝑎𝑥𝑖𝑠.
Magenta Line
f. Reflection in the line 𝑦 = 𝑥, followed by a Reflection in the x-axis.
Orange Line
A. Name the point. then plot its’ reflection over the x axis, y axis, and y=x.
Use the rules above to draw the segments.
𝑥 𝑎𝑥𝑖𝑠 𝑦 𝑎𝑥𝑖𝑠 𝑦 = 𝑥
1. A (3, 2) (𝟑, −𝟐) (−𝟑, 𝟐) (𝟐, 𝟑)
2. B (5, 0) (𝟓, 𝟎) (−𝟓, 𝟎) (𝟎, 𝟓)
3. C (-3, -1) (−𝟑, 𝟏) (𝟑, −𝟏) (−𝟏, −𝟑)
4. Plot each point, then plot its’ reflection in the line 𝑦 = 2. Name the point.
a. M (4, 4) 𝑀′(𝟒, 𝟎)
b. N (-5, 2) 𝑁′(−𝟓, 𝟐)
c. P (-2, -4) 𝑃′(−𝟐, 𝟖)
3. Describe the composition of the transformation using coordinate notation for
the above translations .
a. M (4, 4) (𝒙, 𝒚) → (𝒙, 𝒚 − 𝟐)
b. N (-5, 2) (𝒙, 𝒚) → (𝒙, 𝒚)
c. P (-2, -4) (𝒙, 𝒚) → (𝒙, 𝒚 + 𝟏𝟐)
Rotations Objectives: SWBAT find and rotate images and points over center of rotation
Rotation~ A “turn”
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.