What you'll learn
This topic covers transformations of graphs and functions — how changing a function's equation moves or reshapes its graph. In this guide you will learn the translation, reflection and stretch transformations, the function notation for each, how to describe a transformation fully, and how to sketch the result. These are important higher-tier graph skills.
Key terms and definitions
Transformation — a change that moves or reshapes a graph.
Translation — a slide of the graph (no rotation or resizing).
Reflection — a flip of the graph in an axis.
Stretch — a stretch or squash of the graph in one direction.
Function notation — writing the graph as y = f(x) so changes can be described.
Core concepts
Vertical translations
y = f(x) + a moves the graph up by a (or down if a is negative). The whole graph slides vertically. For example, y = x² + 3 is the graph of y = x² moved up 3. The translation vector is (0, a).
Horizontal translations
y = f(x + a) moves the graph left by a (the change is "inside" the function, so it acts the opposite way). For example, y = (x − 2)² is y = x² moved right 2. The translation vector is (−a, 0). This "inside does the opposite" rule catches many students out.
Reflections
y = −f(x) reflects the graph in the x-axis (the y-values change sign). y = f(−x) reflects it in the y-axis (the x-values change sign). Knowing which sign goes where is essential.
Stretches
y = af(x) is a vertical stretch by scale factor a (y-values multiplied by a). y = f(ax) is a horizontal stretch by scale factor 1/a (x-values divided by a — again the inside acts the opposite way). A factor between 0 and 1 squashes the graph.
Describing transformations fully
To describe a transformation fully, name the type (translation, reflection or stretch) and give the detail — a vector for a translation, the axis for a reflection, or the scale factor and direction for a stretch. Marks require the complete description.
Worked examples
Example 1: Vertical translation
Describe the transformation from y = f(x) to y = f(x) − 4.
A translation by vector (0, −4) — the graph moves down 4.
Example 2: Horizontal translation
The graph y = x² is transformed to y = (x + 3)². Describe it.
A translation by vector (−3, 0) — moved left 3 (inside acts the opposite way).
Example 3: Reflection
Describe the transformation from y = f(x) to y = −f(x).
A reflection in the x-axis.
Common mistakes and how to avoid them
Getting horizontal translations backwards. f(x + a) moves left, not right.
Confusing the two reflections. −f(x) is in the x-axis; f(−x) is in the y-axis.
Mixing up the stretch direction. af(x) stretches vertically; f(ax) stretches horizontally.
Incomplete descriptions. Always give the vector, axis or scale factor.
Wrong scale factor for f(ax). The horizontal stretch factor is 1/a.
Exam technique for Transformations of Graphs
Spot whether the change is inside or outside f( ).
Outside changes affect y (and behave as expected); inside changes affect x (and act the opposite way).
Name the type and give full detail — vector, axis or scale factor.
Track key points (intercepts, turning points) through the transformation.
Sketch the new graph to check it looks right.
Quick revision summary
Transformations move or reshape a graph written as y = f(x). Outside changes affect the y-values and behave as expected: f(x) + a translates up by a, −f(x) reflects in the x-axis, and af(x) is a vertical stretch by factor a. Inside changes affect the x-values and act the opposite way: f(x + a) translates left by a, f(−x) reflects in the y-axis, and f(ax) is a horizontal stretch by factor 1/a. To describe a transformation fully, name the type and give the detail — a vector for a translation, the axis for a reflection, the scale factor and direction for a stretch. The common errors are reversing horizontal translations, confusing the two reflections, mixing up stretch directions, and giving incomplete descriptions. Decide whether the change is inside or outside f( ), apply the right rule, track key points, and describe the transformation completely.