1. Composite Transformations
Allison Wai & Manizeh Ahmed

2. Composite Transformation is…
A sequence of various transformations, such as
Translation
Reflection
Rotation
Scaling/Homothety (Dilatation)

3. Translation A’ (-3, 4) B’ (1, -1) C’ (-5, -4) D’ (1, -6)
The task of moving coordinates on a graph, while maintaining the shape of a figure. Rule: t (x, y) → (x+h, y+k) Ex. t (x, y) = (x+2, y–1) A (-5, 5) B (-1, 0) C (-7, -3) D (-1, -5) A A’
A’ (-3, 4)
B’ (1, -1)
C’ (-5, -4)
D’ (1, -6) B B’ C D C’ D’

4. Rotation r 90°(x, y) → (-y, x) r 180° (x, y) → (-x, -y)
Rotating a figure by a set number of degrees. Rule: Ex. r 270° (x, y) → (y, -x) A (-5, 5) B (-1, 0) C (-7, -3) D (-1, -5)
r 90°(x, y) → (-y, x)
r 180° (x, y) → (-x, -y)
r 270° (x, y) → (y, -x)
r 360° (x, y) → (x, y) C’ A A’ D’ B’ B
A’ (5, 5)
B’ (0, 1)
C’ (-3, 7)
D’ (-5, 1) C D

5. Reflection Sx (x, y) → (x, -y) Sy (x, y) → (-x, y)
Reflecting the coordinates of a figure across the x, or y axis. Rule: Ex. S2nd bisector (x, y) → (-y, -x) A (-5, 5) B (-1, 0) C (-7, -3) D (-1, -5)
Sx (x, y) → (x, -y)
Sy (x, y) → (-x, y)
Diagonals
S1st bisector (x, y) → (y, x)
S2nd bisector (x, y) → (-y, -x) C’ A’ A B’ D’ B
A’ (-5, 5)
B’ (0, 1)
C’ (3, 7)
D’ (5, 1) C D

6. scaling eh (x, y) → (kx, y) ev (x, y) → (x, ky)
Increasing or decreasing the size of a figure vertically or horizontally
Rule:
Ex. ev (x, y) → (x, 2y)
A (-5, 5)
B (-1, 0)
C (-7, -3)
D (-1, -5)
Scale factor is ‘k’
Horizontal (x axis)
eh (x, y) → (kx, y)
Vertical (y axis)
ev (x, y) → (x, ky) A’ C’ A B’ B C
A’ (-5, 10)
B’ (-1, 0)
C’ (-7, 6)
D’ (-1, -10) D D’

7. homothety (dilatation)
Increasing or decreasing the size of a figure both vertically and horizontally Rule: Ex. h (x, y) → (2x, 2y) A (-5, 5) B (-1, 0) C (-4, -3) D (-1, -5)
h (x, y) → (kx, ky) A’ A B
A’ (-10, 10)
B’ (-2, 0)
C’ (-8, -6)
D’ (-2, -10) B’ C D C’ D’

8. Composite transformation
A series of transformations— a combination of translation, rotation, reflection, scaling or homothety (dilatation)— performed consecutively. With each step, performing a transformation results in a new set of points. ↓ ↓ ↓ ↓ ↓ ↓ The new set of points is now what the next transformation will be based on. Ex. Start with a translation: t (2, -1) A (-5, 5) B (-1, 0) C (-7, -3) D (-1, -5)
Next, rotation: r 180° (x, y) → (-x, -y)
Original set of points
A’ (-3, 4)
B’ (1, -1)
C’ (-5, -4)
D’ (1, -6)
A’’ (3, -4)
B’’ (-1, 1)
C’’ (5, 4)
D’’ (-1, 6)
And so on...
Now use the new set of points to continue

9. Remember! Sx ° h (k, 2) ° r -270° ° t (-2,-3) Work backwards!
When given a formula written like this:
Sx ° h (k, 2) ° r -270° ° t (-2,-3)
Work backwards!