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Composite Transformations

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Presentation on theme: "Composite Transformations"— Presentation transcript:

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!


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