Transformations

Understanding Transformations

• A transformation applies to the change in position, shape, or size of a graph.

• You need to be able to recognise four main types of transformations: translation, reflection, stretch, and rotation.

• Each of these transformations has a predictable effect on the graph of a function and can be described using mathematical language.

Translations

• A translation moves a shape up, down, left or right, but it does not change its size, orientation or shape.

• In terms of a graph, it moves each point to a new location, a fixed distance in a given direction.

• When a function is translated, it results in a horizontal or vertical shift. The graph of the function retains the exact same shape and orientation - it is just relocated.

• For example, the graph of y = f(x) + c is a vertical translation of the graph of y = f(x) by ‘c’ units. Similarly, y = f(x + c) is a horizontal translation of the graph of y = f(x) by ‘-c’ units.

Reflections

• A reflection flips a graph about a line without changing its size or shape.

• This reflection line is normally the x-axis (y = f(-x)) or the y-axis (y = -f(x)), but could also be another line (e.g., y = x).

• The graph appears to be reflected in the chosen line of reflection.

Stretches

• A stretch pulls a graph vertically or horizontally, distorting its original size but keeping the same shape.

• Vertical stretch is represented by y = af(x), where ‘a’ is the scale factor. If

  a

  > 1, it’s a stretch, if 0 <

  a

  < 1, it’s a compression.

• Horizontal stretch is represented by y = f(bx), where ‘b’ is the scale factor. Here, the scale factor is the reciprocal of ‘b’; hence if

  b

  > 1, it’s a compression, and if 0 <

  b

  < 1, it’s a stretch.

Rotations

• Rotations in transformations are less common in A-Level Pure Mathematics but still important to understand.

• It involves turning a graph around a single point, known as the centre of rotation.

• The graph will maintain its shape and size, but its orientation and position will change.

Finally, keep in mind that you might be asked to combine transformations or perform the inverse of a transformation. Understanding each transformation thoroughly will help with these more complex tasks.