IGCSE - Extended Mathematics Transformation

IGCSE - Extended MathematicsTransformationContent: Reflection (M): Reflect simple plane figures in horizontal or vertical lines; Rotation (R): Rotate simple plane figures about the origin, vertices or midpoints of edges of the figures,through multiples of 90 ; Enlargement (E) : Construct given translations and enlargements of simple plane figures; Shear (H) &Stretch (S) Their combinations (if M(a) b and R(b) c the notation RM(a) c will be used; Invariants under these transformations may be assumed.) Identify and give precise descriptions oftransformations connecting given figures. Describe transformations using co-ordinates and matrices (singular matrices are excluded).Transformation: The word” transform “means "to change." In geometry, a transformation changes the position of a shape on acoordinate plane. That means a shape is moving from one place to another. The original shape of the object is called the pre-image and the final shape and position of the object is the imageunder the transformation.Isometry : Isometric transformation is a transformation that preserves congruence. In other words, a transformation inwhich the image and pre-image have the same side lengths and angle measurements. The following transformationsmaintain their mathematical congruence Reflection (Flip) ,Translation (Slide), Rotation (Turn).Mathelaureate / IGCSE / Transformation / D.JPage no: 1

TransformationEffect ( Image )Matrix1. IdentityMatrixExampleNo Effect , Image Remains SameThis transformation matrix is the identitymatrix. When multiplying by this matrix, the point matrix is unaffected and the new matrixis exactly the same as the point matrix.2. ReflectionReflection in the X axisThis transformation matrix creates areflection in the x-axis. When multiplying bythis matrix, the x co-ordinate remainsunchanged, but the y co-ordinate changessign.Mathelaureate / IGCSE / Transformation / D.JPage no: 2

3. ReflectionReflection in the Y axisThis transformation matrix creates areflection in the y-axis. When multiplying bythis matrix, the y co-ordiante remainsunchanged, but the x co-ordinate changessign.4. Rotation 180 degrees RotationThis transformation matrix creates a rotationof 180 degrees. When multiplying by thismatrix, the point matrix is rotated 180degrees around (0,0). This changes the sign ofboth the x and y co-ordinates.5. Reflection Reflection on y x lineThis transformation matrix creates areflection in the line y x. When multiplying bythis matrix, the x co-ordinate becomes the yco-ordinate and the y-ordinate becomes the xco-ordinate.Mathelaureate / IGCSE / Transformation / D.JPage no: 3

6. Rotation90 degrees clock wise rotationThis transformation matrix rotates the pointmatrix 90 degrees clockwise. Whenmultiplying by this matrix, the point matrix isrotated 90 degrees clockwise around (0,0).7. Rotation Anti-clockwise rotation 90 degreesThis transformation matrix rotates the pointmatrix 90 degrees anti-clockwise. Whenmultiplying by this matrix, the point matrix isrotated 90 degrees anti-clockwise around(0,0).8. Reflection Reflection in the line y -xThis transformation matrix creates areflection in the line y -x. When multiplyingby this matrix, the point matrix is reflected inthe line y -x changing the signs of both coordinates and swapping their values.Mathelaureate / IGCSE / Transformation / D.J Page no: 4

9 .EnlargementEnlargement of scale factorThis transformation matrix is the identitymatrix multiplied by the scalar 6. Whenmultiplying by this matrix, the point matrix isenlarged by a factor of 6 in the x and ydirections. 10.EnlargmentThis transformation matrix is the identitymatrix but T1,1has been enlarged by a factor of7 and T2,2 has been enlarged by a factor of 0.When multiplying by this matrix, the x coordinate is enlarged by a factor of 7, whilstthe y co-ordinate is enlarged by a factor of 0. This transformation matrix is the identitymatrix but T1,1has been enlarged by a factor ofa and T2,2 has been enlarged by a factor of b.When multiplying by this matrix, the x coordinate is enlarged by a factor of a, whilstthe y co-ordinate is enlarged by a factor of b. 11. Enlargment12. EnlargmentMathelaureate / IGCSE / Transformation / D.JPage no: 5

This transformation matrix creates a rotationand an enlargement. When multiplying by thismatrix, the point matrix is rotated 90 degreesanticlockwise around (0,0), whilst the x xoordinate of the new point matrix is enlargedby a factor of -5 and the y co-ordinate of thenew point matrix is enlarged by a factor of 7.13. Streatch Horizontal axis stretch k 0 0 1 Stretch, scale factor k parallel to the x- axisMathelaureate / IGCSE / Transformation / D.JPage no: 6

13. StreatchVertical axis stretch 1 0 0 k Stretch, scale factor k parallel to the y-axis14. ShearShear parallel to the x-axis 1 k 0 1 The matrix corresponds to a shear parallel tothe x-axis. Points on the x-axis do not move,whilst points on the line y 1 are translated kunits to the right.Mathelaureate / IGCSE / Transformation / D.JPage no: 7

15. ShearShear parallel to y axis 1 0 k 1 The matrix corresponds to a shear parallel tothe y-axis. Points on the y-axis do not move,whilst points on the line x 1 are translated kunits up.Mathelaureate / IGCSE / Transformation / D.JPage no: 8

Mathelaureate / IGCSE / Transformation / D.J Page no: 1 IGCSE - Extended Mathematics Transformation Content: Reflection (M): Reflect simple plane figures in horizontal or vertical lines; Rotation (R): Rotate simple plane figures about the origin, vertices or midpoints of edges of the .