How well do you know your transformations?

How well do you know your transformations?

Understanding Transformations

  • Transformations apply to any point on a graph, altering its position or orientation.
  • Basic transformations include translation, reflection, rotation, and stretching or compression.
  • More complex transformations can be made by combining these basic ones.

Translation

  • Translation moves each point a consistent distance in a specified direction.
  • In terms of graphs, the point (x, y) translated by the vector for t units is denoted as (x+t,y).
  • Likewise, if a point is translated upwards/downwards, the point (x, y) is transferred to (x, y+t).

Reflection

  • Reflecting a graph gives a mirror image about a certain line known as the line of reflection.
  • A reflection in the line y = x transforms the point (x, y) into (y, x).
  • Similarly, a reflection in the y-axis transforms the point (x, y) into (-x, y).

Rotation

  • Rotating a graph spins it around a fixed point, known as the centre of rotation.
  • The most common rotations are 90°, 180°, or 270° both clockwise and anticlockwise.
  • For rotation 90° anti-clockwise about the origin, the point (x,y) maps to (-y,x).

Stretching and Compression

  • Stretching pulls a graph away from an axis, whilst compression pushes it towards an axis.
  • If we stretch a graph parallel to the x-axis by a scale factor of k from the line y = a, the point (x, y) maps to (kx, y).
  • Compression works in the same way except it pushes the graph towards an axis.

Combining Transformations

  • A single transformation can be achieved by combining two or more simple transformations.
  • The order in which transformations are applied can affect the final result; hence, it’s important to follow the specified order.

Using Transformation Matrices

  • Transformation matrices are a handy tool for performing geometric transformations, particularly for rotation and reflection.
  • For 2D transformations, matrices are 2x2, and 3x3 for 3D.
  • To apply a transformation, multiply the transformation matrix with the coordinates of each relevant point.

Improving Transformation Skills

  • Practice is essential for improving transformation skills. Start with simple transformations and gradually add complexity.
  • Sketching helps visualize transformations, particularly for rotations and reflections.
  • Mastering matrix manipulation helps with the application of transformations, especially when dealing with 3D shapes.