2. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Do Now: Reflect ΔCDE through the line y = -2. Aim: How do we move from here to there, mathematically speaking?

3. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Translation * also called a “slide” Movement of an object a specified distance, in a specified direction, along a straight line.

4. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Translation Properties: Distance is preserved Angle Measure is preserved Parallelism is preserved Collinearity is preserved A midpoint is preserved Area is preserved

5. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Translation in Coordinate Geometry Translate point P(1, 3) +3 units in the x direction and –2 units in the y direction.Notation - (T 3,-2 ). P(1, 3) +3 –2 P’(4, 1) Under T 3,-2 P(1, 3)  P’(4, 1) P(x, y)  P’(x + a, y + b) a b Under a translation of a units in the horizontal (x) direction and b units in the vertical (y) direction, the image of P(x, y) is P’(x + a, y + b). General Rule:

6. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Model Problem: Translation Under a translation in the diagram below, the image of A becomes H. Under the same translation, find the image for: B FH I ACBDE F GHIJ KLMNO

7. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Model Problems: Translation Find the coordinates of the image of the point (-4, 6) under T 4,-1 Which translation maps (5, -3)  (3, 5) 1) T 2, 2 2) T -2, 8 3) T -2, 2 If a translation T 3, 1 maps (4, 2) onto (x, 3), what is the value of x? (0, 5) 2 7 P(x, y)  P’(x + a, y + b)

8. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Dilation x 2 x 4 A DILATION is a reduction or enlargement of a figure by a given scale factor, from a given point. x 2 x 4 x 8 Given Point x 16 scale factor

9. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Dilation Properties Properties: Distance is preserved Angle Measure is preserved Parallelism is preserved Collinearity is preserved A midpoint is preserved Area is preserved

10. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Dilation in Coordinate Geometry Notation for DILATION: D x ( x is the scale factor) Dilate  ABC by a factor of 3 from the origin - D 3 from origin. (1,1)(3,1) (2,3) AB C (3,3) (6,9) (9,3) A’ B’ C’ Multiply both the x and y coordinate by the scale factor to find the new point P(x, y)  P’(kx, ky) or D k (x, y) = (kx, ky)

11. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Model Problems: Dilation Find the image of each of the following points under the dilation D 6 1)(3, 5) 2)(-4, 1) 3)(-6, -8) Find the image of each of the following points under the dilation D 1/2 1)(3, 5) 2)(-4, 1) 3)(-6, -8) (18, 30) (-24, 6) (-36, -48) (1.5, 2.5) (-2,.5) (-3, -4) Does the transformation (3, 1)  (12, 1) represent a dilation D 4 ? Explain No. Only the x coordinate was dilated by the scale factor

12. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Rotation A The turning of a figure about some fixed point (A). The turn is measured in degrees

13. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Model Problems: Rotation The black lines in the figure below intersect and form twelve 30 0 angles. A J B C K D E L F G H M I O

14. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Rotation Properties Properties: Distance is preserved Angle Measure is preserved Parallelism is preserved Collinearity is preserved A midpoint is preserved Area is preserved

15. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Rotation in Coordinate Geometry Rotate  ABC about the original Notation - Rotate  ABC about the original Notation - A B C B’ A’ C’ Rotate of 90 0 about the origin - R 90º (x,y) = (y,-x) Rotate of 180 0 about the origin - R 180º (x,y) = (-x,-y) Rotate of 270 0 about the origin - R 270º (x,y) = (-y,-x) (-5,1) (-5,4) (-2,1) (4,5) (1,5) (1,2) (5,-1) A” (5,-4)B” (2,-1) C” (-1,-5) A’” (-4,-5) B’” (-1,-2)C’”

16. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Transformations Rotate of 90 0 about the origin - R 90º (x,y) = (y,-x) Rotate of 180 0 about the origin - R 180º (x,y) = (-x,-y) Rotate of 270 0 about the origin - R 270º (x,y) = (-y,-x) Under reflection in the y-axis (r y ), the image of P(x, y) is P’(-x, y) - Under reflection in the x-axis, the image of P(x, y) is P”(x, -y) - Under reflection in the y-axis (r y ), the image of P(x, y) is P’(-x, y) - Under reflection in the x-axis, the image of P(x, y) is P”(x, -y) - Reflection (r): Translation (T): Under a translation of a units in the horizontal (x) direction and b units in the vertical (y) direction, the image of P(x, y) is P’(x + a, y + b). T a,b (x,y) = (x + a, y + b) Rotation (R): D k (x,y) = (kx, ky) - (k is the scale factor, k ≠ 0) Dilation (D): r y-axis (x,y) = (-x,y) r x-axis (x,y) = (x,-y)

17. Aim: Transformation: Translation, Rotation, Dilation Course: Alg. 2 & Trig. Preserved Properties Line Reflection Point Reflection RotationTranslationDilation Distance  Measure Parallelism Collinearity Midpoint Betweeness of points Area NOYYY YYYYY YYYYY YYYYY YYYYY YYYYY Y YYY Y