1. WARM UP:
Draw pentagon PENTA on three different graphs on your worksheet. Label the vertices and write each vertex as an ordered pair.
On the first graph, TRANSLATE pentagon PENTA five units right and 2 units down. Label the vertices of P´E´N´T´A´ and write each vertex as an ordered pair.
On the second graph, REFLECT pentagon PENTA across the line 𝑦=3. Label the vertices of P´E´N´T´A´ and write each vertex as an ordered pair.
On the third graph, ROTATE pentagon PENTA 90° around the origin. Label the vertices of P´E´N´T´A´ and write each vertex as an ordered pair. E P N A T

2. Algebraic Representations of Transformations
Essential Question?
How can you describe the effect of a translation, rotation, or reflections on coordinates using an algebraic representation?
8.G.3

3. Common Core Standard:
8.G ─Understand congruence and similarity using physical models, transparencies, or geometry software.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

4. Objectives:
To describe the effect of a translation, rotation, or reflections on coordinates using an algebraic representation.

5. Algebraic Representations of Translations
Representing a TRANSLATION algebraically is rather easy. RIGHT/UP means POSITIVE (ADD) LEFT/DOWN means NEGATIVE (SUBTRACT) The algebraic notation looks like this: 𝒙,𝒚 →(𝒙±𝒂, 𝒚±𝒃) or 𝒙,𝒚 → 𝒙±𝒂, 𝒚±𝒃

6. Algebraic Representations of Translations
How can we represent translations using algebraic notation?
TRANSLATIONS:
Move right
𝒙,𝒚 →(𝒙+𝒂, 𝒚)
Move left
𝒙,𝒚 →(𝒙−𝒂, 𝒚)
Move up
𝒙,𝒚 →(𝒙, 𝒚+𝒃)
Move down
𝒙,𝒚 →(𝒙, 𝒚−𝒃)

7. Algebraic Representations of Translations
Let’s see what this means: Translate the following points using the rule: 𝒙,𝒚 →(𝒙+𝟑, 𝒚−𝟐)
Preimage
Image A
(2, 3) A´ B
(−3, 5) B´ C
(7, −1) C´ D
(−4, −5) D´

8. Try this one:
Translate the following points using the rule: 𝒙,𝒚 → 𝒙−𝟕, 𝒚+𝟒
Preimage
Image A
(−12, 5) A´ B
(−2, −11) B´ C
(7, 1) C´ D
(−9, 7) D´

9. Graphing an image after a transformation
Step 1: Create a table for each of the vertices. Step 2: Graph the new points (the primes) Step 3: Connect the line segments (using a ruler)
Preimage
Image A A´ B B´ C C´ D D´

10. Example: Preimage Image X (0,0) X´ Y (2,3) Y´ Z (4,−1) Z´
Triangle XYZ, with vertices X(0,0), Y(2,3), and Z(4,−1) undergoes a translation of 3 units to the right and 1 unit down. Write the translation using algebraic notation, find the vertices of X´ Y´ Z´, then graph the triangle and its image. Algebraic notation:
Preimage
Image X
(0,0) X´ Y
(2,3) Y´ Z
(4,−1) Z´

11. Example: Algebraic notation: 𝒙,𝒚 →(𝒙+𝟑, 𝒚−𝟏) Preimage Image X (0, 0)
(3,−1) Y
(2, 3) Y´
(5, 2) Z
(4,−1) Z´
(7,−2)

12. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across the x-axis. A B C B´ A´ C´
When a figure is reflected across the x-axis, the x-coordinates stay the same and the y-coordinates are negated.
𝒙,𝒚 → 𝒙,−𝒚

13. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across the y-axis. A B C B´ A´ C´
When a figure is reflected across the y-axis, the y-coordinates stay the same and the x-coordinates are negated.
𝒙,𝒚 → −𝒙,𝒚

14. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across any horizontal line. A B C B´ C´ A´
When a figure is reflected across any horizontal line the x-values still stay the same. The y-values are negated and then shifted by TWICE the value of the line.
𝒙,𝒚 → 𝒙,−𝒚+𝟐𝒃

15. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across any vertical line. A B´ A´ C´ B C
When a figure is reflected across any vertical line the y-values still stay the same. The x-values are negated and then shifted by TWICE the value of the line.
𝒙,𝒚 → −𝒙+𝟐𝒂,𝒚

16. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across the oblique line 𝑦=𝑥. A B C B´ A´ C´
When a figure is reflected across the line 𝑦=𝑥
𝒙,𝒚 → 𝒚,𝒙

17. Algebraic Representations of Reflections
Think about what happens when a figure is reflected across the oblique line 𝑦=−𝑥. A C B B´ A´ C´
When a figure is reflected across the line 𝑦=−𝑥
𝒙,𝒚 → −𝒚,−𝒙

18. Algebraic Representations of Reflections
How can we represent a reflection using algebraic notation?
REFLECTIONS:
Across the x-axis
Multiply the y-coordinate by −1 𝒙,𝒚 →(𝒙, −𝒚)
Across any horizontal line
𝑦=𝑏
𝒙,𝒚 →(𝒙, −𝒚+𝟐𝒃)
Across the y-axis
Multiply the x-coordinate by −1 𝒙,𝒚 →(−𝒙, 𝒚)
Across any vertical line
𝑥=𝑎
𝒙,𝒚 →(−𝒙+𝟐𝒂, 𝒚)
Across 𝑦=𝑥
𝒙,𝒚 →(𝒚, 𝒙)
Across 𝑦=−𝑥
𝒙,𝒚 →(−𝒚, −𝒙)

19. Example: Preimage Rule: Image X (0,0) X´ Y (2,3) Y´ Z (4,−1) Z´
Triangle XYZ, with vertices X(0,0), Y(2,3), and Z(4,−1) undergoes a reflection across the x-axis. Write the reflection using algebraic notation, find the vertices of X´ Y´ Z´, and graph the triangle and its image. Algebraic Notation:
Preimage
Rule:
Image X
(0,0) X´ Y
(2,3) Y´ Z
(4,−1) Z´
(𝒙,𝒚) → (𝒙, −𝒚)

20. Example: Algebraic notation: 𝒙,𝒚 →(𝒙, −𝒚) Preimage Image X (0, 0) X´ Y
(2, 3) Y´
(2,−3) Z
(4,−1) Z´
(4, 1)

21. Algebraic Representations of Rotations only if the center of rotation is the origin
Think about what happens when a figure is rotated 90° clockwise about the origin. A B C B´ A´ C´
Take B and B´ for example:
B(9,3) B´(3,−9)
What about A and A´:
A(3,7) A´(7,−3)
Can you think of a rule?
Multiply the x-coordinate by −1, then switch x & y
𝒙,𝒚 →(𝒚, −𝒙)

22. Algebraic Representations of Rotations only if the center of rotation is the origin
What happens when a figure is rotated 90° counterclockwise about the origin? A B C B´ A´ C´
Take B and B´ for example:
B(9,3) B´(−3,9)
What about A and A´:
A(3,7) A´(− 7,3)
Can you think of a rule?
Multiply the y-coordinate by −1, then switch x & y
𝒙,𝒚 →(−𝒚,𝒙)

23. Algebraic Representations of Rotations only if the center of rotation is the origin
What happens when a figure is rotated 180° about the origin? A B C B´ A´ C´
Take B and B´ for example:
B(9,3) B´(−9,−3)
What about A and A´:
A(3,7) A´(−3,−7)
Can you think of a rule?
Multiply both the x and y-coordinate by −1
𝒙,𝒚 →(−𝒙, −𝒚)

24. Algebraic Representations of Rotations only if the center of rotation is the origin
How can we represent a rotation using algebraic notation?
ROTATIONS:
90° clockwise
Multiply the x-coordinate by −1, then switch the x- and y- coordinates:
𝒙,𝒚 →(𝒚, −𝒙)
90° counterclockwise
Multiply the y-coordinate by −1, then switch the x- and y- coordinates:
𝒙,𝒚 →(−𝒚,𝒙)
180°
Multiply the both coordinates by −1:
𝒙,𝒚 →(−𝒙,−𝒚)

25. Example: Preimage Rule: Image A (−4,2) A´ B (−3,4) B´ C (2,3) C´ D
Quadrilateral ABCD has vertices at A(−4,2), B(−3,4), C(2,3), and D(0,0). Find the vertices of quadrilateral A´B´C´D´ after a 90° clockwise rotation. Then graph the quadrilateral and its image. Algebraic Notation:
Preimage
Rule:
Image A
(−4,2) A´ B
(−3,4) B´ C
(2,3) C´ D
(0,0) D´
(𝒙,𝒚) → (𝒚, −𝒙)

26. Example: Algebraic notation: 𝒙,𝒚 →(𝒚, −𝒙) Preimage Image A (−4, 2) A´
(2, 4) B
(−3, 4) B´
(4, 3) C
(2, 3) C´
(3,−2) D
(0, 0) D´