Key Terms
- Translation: Sliding the graph up, down, left, or right without changing its shape
- Reflection: Flipping the graph in an axis
- Stretch: Enlarging or squashing the graph in one direction
- f(x) notation: A way of describing a function so transformations can be written generally
- Invariant point: A point on the graph that does not move after a transformation
Must-Know Facts
- f(x) + a: translate UP by a (outside the bracket = y-direction)
- f(x + a): translate LEFT by a ā inside bracket, OPPOSITE direction
- af(x): vertical stretch by factor a (y-coordinates multiplied by a)
- f(ax): horizontal squash by factor a (x-coordinates divided by a)
- āf(x): reflection in the x-axis (y-values change sign)
- f(āx): reflection in the y-axis (x-values change sign)
- Inside the bracket affects x in the OPPOSITE direction to what you expect
Key Methods
- To find new coordinates: apply each transformation to the x and y values in turn
- Vertical transformations (outside bracket) only change y-coordinates
- Horizontal transformations (inside bracket) only change x-coordinates
- For combined transformations, apply them one at a time in the order given
Key Formulas
- f(x) + a: translate up by a (y changes, x stays)
- f(x + a): translate left by a (x changes in opposite direction)
- af(x): stretch vertically by factor a (y values multiply by a)
- f(ax): stretch horizontally by factor 1/a (x values divide by a)
- āf(x): reflect in x-axis; f(āx): reflect in y-axis
Common Mistakes
- f(x + a) shifts left not right: Inside the bracket, the direction is OPPOSITE ā f(x + 2) shifts LEFT by 2
- f(ax) stretches by 1/a not a: f(2x) makes the graph NARROWER (horizontal stretch by Ā½)
- Applying transformations in wrong order: For combined transformations, apply them in the stated order ā order matters
- Confusing vertical and horizontal stretches: Outside bracket = vertical (affects y); inside bracket = horizontal (affects x)
This topic summary covers Knowledge Organiser: Graph Transformations within Graph Transformations for GCSE Mathematics. Revise Graph Transformations in Graphs for GCSE Mathematics with 14 exam-style questions and 1 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 4 of 4 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Graph Transformations
The graph of y = f(x) is transformed to y = f(x) + 3. Which of the following describes this transformation?
Describe fully each of the following transformations of y = f(x): (a) y = 3f(x) (b) y = f(2x) (c) y = f(x) - 5
Quick recall flashcards
f(x + a) vs f(x) + a
f(x + a) = LEFT by a (inside, opposite). f(x) + a = UP by a (outside, same direction)