Graph transformations let you shift, flip, and stretch the graph of any function without recalculating every coordinate. At GCSE, you need four types — translation, reflection, and two kinds of stretch — and you describe them precisely using f(x) notation. Knowing the rules means you can sketch transformed curves quickly in an exam.
What is f(x) notation and why does it matter?
When a graph is described as y = f(x), the letter f stands for whatever function produces the curve. Using this notation lets us describe any transformation in a shorthand that works for any shape — parabola, cubic, circle, or trigonometric curve.
The key distinction you must understand is:
- Changes outside the function — like f(x) + 3 — act on the y-values, so they shift or scale the graph vertically.
- Changes inside the function — like f(x + 3) — act on the x-values, so they shift or scale the graph horizontally. Crucially, they act in the opposite direction to what you might expect.
This "inside vs outside" rule is the foundation of all four transformations.
What are the four graph transformations at GCSE?
Here is a summary of every transformation you need, with how each affects a general graph y = f(x).
| Transformation | New equation | Effect on graph |
|---|---|---|
| Vertical translation up by a | y = f(x) + a | Every point moves up by a units |
| Vertical translation down by a | y = f(x) − a | Every point moves down by a units |
| Horizontal translation right by a | y = f(x − a) | Every point moves right by a units |
| Horizontal translation left by a | y = f(x + a) | Every point moves left by a units |
| Reflection in the x-axis | y = −f(x) | Every y-coordinate changes sign |
| Reflection in the y-axis | y = f(−x) | Every x-coordinate changes sign |
| Vertical stretch, scale factor a | y = af(x) | y-coordinates multiplied by a |
| Horizontal stretch, scale factor 1/a | y = f(ax) | x-coordinates divided by a |
The most commonly confused pairs are f(x + a) (left, not right) and f(ax) (horizontal stretch of factor 1/a, not a). Fixing those two in your mind will prevent the most common exam errors.
How do translations work — with a worked example?
A translation slides the graph without changing its shape or orientation. It is described by a column vector.
Worked example: The graph of y = f(x) is shown. Sketch y = f(x − 2) + 3.
- The "+3" is outside f, so move every point up 3 units.
- The "−2" is inside f (replacing x with x − 2), so move every point right 2 units.
- The combined transformation is a translation by the vector $\binom{2}{3}$.
A key point: if the original graph crosses the x-axis at (1, 0), the translated graph crosses at (3, 3). Apply the same shift to every labelled point — the shape is identical.
How do reflections work?
Reflections flip the graph about an axis.
- y = −f(x): multiply every y-coordinate by −1. The graph flips over the x-axis. A maximum point becomes a minimum.
- y = f(−x): replace every x-coordinate with its negative. The graph flips over the y-axis.
Worked example: y = f(x) has a maximum at (3, 5). After the transformation y = −f(x), where is the maximum?
The x-coordinate stays at 3. The y-coordinate becomes −5. So there is now a minimum at (3, −5).
Note: for an even function (symmetric about the y-axis), y = f(−x) looks exactly the same as y = f(x).
How do stretches work — and what trips students up?
A vertical stretch by scale factor a multiplies every y-coordinate by a: $$y = af(x)$$
A horizontal stretch by scale factor a divides every x-coordinate by a: $$y = f(ax)$$
The trap with horizontal stretches: f(2x) stretches horizontally by factor ½, not 2. This is because any x-value that previously produced a given y-value now needs to be half as large for the argument to reach the same point in f.
Worked example: y = f(x) passes through (6, 4). After y = f(3x), what point does the graph pass through?
For f(3x), x-coordinates are divided by 3 (horizontal stretch factor ½ × 2 = ... more precisely, factor 1/3). New point: (2, 4). The y-coordinate is unchanged.
How do you identify a transformation from a sketch?
- Compare key points. Pick a labelled maximum, minimum, or intercept on the original and find where it ended up.
- Check if the shape stretched or moved. If every point shifted the same amount, it is a translation. If y-values scaled, it is a vertical stretch or reflection.
- Write the vector or equation. A translation of left 4 and up 1 is y = f(x + 4) + 1. A vertical stretch by factor 3 is y = 3f(x).
GCSE mark schemes require you to state the transformation type AND its full description (vector for translations, scale factor for stretches, axis for reflections).
Which transformations appear most often in GCSE exams?
Translations and reflections appear more frequently than stretches at GCSE. The question types to practise are:
- Sketch the transformed graph (draw the new curve with key points labelled).
- Write the equation of the transformation shown.
- State the transformation that maps one curve onto another.
Trigonometric graphs (y = sin x, y = cos x) appear with translations and stretches: for example, y = sin(2x) is a horizontal compression of y = sin x by factor ½, giving double the frequency.
Frequently asked questions
Why does f(x + 3) move the graph to the LEFT, not the right?
Think of it this way: for the original graph y = f(x), the y-value at x = 5 becomes, in the new graph, the y-value at x = 2 (because f(2 + 3) = f(5)). So the point x = 5 from the original now appears at x = 2 — the whole graph has shifted left. The rule is: adding a positive number inside the brackets shifts the graph in the negative x-direction.
Does the order of transformations matter?
Yes — when a graph is subject to more than one transformation and they involve the same variable, order can matter. For horizontal transformations, apply any stretch before any translation if they both act inside f. For unambiguous exam questions involving combinations in the standard forms above, applying translations and stretches as described will always give the correct result.
How do I remember which transformation each form gives?
Use the "inside vs outside" rule as your anchor. Outside changes → vertical effects (obvious). Inside changes → horizontal effects (opposite to what you expect). Then check sign: f(x − a) is right (not left), f(ax) is squash (not stretch). Practise labelling the coordinates of three or four key points after each transformation until it is automatic.
What is the difference between a stretch and an enlargement?
A stretch in graph transformations scales the graph in only one direction — either vertically or horizontally — without changing the other. An enlargement (from geometry) scales equally in all directions from a centre point. At GCSE these are completely separate topics; stretches of graphs never count as enlargements.
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