Graph transformations are worth 4–6 marks on most higher tier papers, and they appear on AQA (8300), Edexcel (1MA1), and OCR (J560) almost every year. The logic is elegant once you see it — a small change inside or outside f(x) moves or stretches the whole curve in a predictable way. This guide covers all four transformation types you need to know, with exactly the kind of worked examples you will meet in the exam.
What You Need to Know First
- You must be able to read and plot coordinates accurately.
- Know what basic graphs look like: y = x², y = x³, y = sin x, y = cos x, y = 1/x.
- Understand function notation — f(x) is the original curve. Any change to f(x) transforms it.
- The transformations only appear on the higher tier paper for all three boards.
- Transformations can be combined — you may be asked to apply two in sequence.
The Four Transformation Types
There are four transformations you need to know, and they all follow from a small set of rules. The key insight: changes inside the bracket affect x (horizontal) and behave counter-intuitively; changes outside the bracket affect y (vertical) and behave as you would expect.
Type 1: Translation — f(x) + a
Rule: y = f(x) + a moves the graph up by a (or down if a is negative).
The change is outside the brackets, so it affects the y-values directly. Every point on the curve shifts vertically by a.
Worked Example 1
The graph of y = f(x) passes through the point (2, 5). Write down the coordinates of this point on the graph y = f(x) + 3.
Every y-coordinate increases by 3. So (2, 5) maps to (2, 8).
The x-coordinate is unchanged because the transformation only affects the output of f(x), not the input.
Type 2: Translation — f(x + a)
Rule: y = f(x + a) moves the graph left by a (counter-intuitive — adding a inside moves it left; subtracting a inside moves it right).
Worked Example 2
The graph of y = f(x) has a maximum at (4, 7). State the coordinates of the maximum of y = f(x − 2).
Here a = −2 inside the bracket, so the graph shifts right by 2. The x-coordinate increases by 2, the y-coordinate stays the same.
Maximum moves to (6, 7).
Memory trick: Think of it this way — if f(x − 2) = 0 needs solving, you set x − 2 equal to what x was for f(x) = 0. So x is 2 bigger than before. The curve shifts right.
Type 3: Reflection — f(−x) and −f(x)
Two separate reflections to know:
- y = f(−x): reflect in the y-axis (flip left–right). The x-coordinates change sign; y-coordinates stay the same.
- y = −f(x): reflect in the x-axis (flip up–down). The y-coordinates change sign; x-coordinates stay the same.
Worked Example 3
The graph y = f(x) passes through the points (−3, 2), (0, 5) and (4, −1).
(a) Write down the image of each point under y = f(−x).
Reflect in the y-axis — negate the x-coordinates:
(−3, 2) → (3, 2)
(0, 5) → (0, 5) (the y-axis point stays put)
(4, −1) → (−4, −1)
(b) Write down the image of each point under y = −f(x).
Reflect in the x-axis — negate the y-coordinates:
(−3, 2) → (−3, −2)
(0, 5) → (0, −5)
(4, −1) → (4, 1)
Type 4: Stretch — af(x) and f(ax)
Two stretch types to know:
- y = af(x): stretch parallel to the y-axis by scale factor a. Multiply all y-coordinates by a; x-coordinates unchanged.
- y = f(ax): stretch parallel to the x-axis by scale factor 1/a. Divide all x-coordinates by a; y-coordinates unchanged.
Again, the counter-intuitive rule applies to the horizontal stretch: multiplying x by a inside the bracket actually squashes the graph (scale factor 1/a), not stretches it.
Worked Example 4
The curve y = f(x) has a turning point at (6, 4).
(a) Give the coordinates of the turning point of y = 3f(x).
Stretch parallel to y-axis, scale factor 3 — multiply y by 3:
Turning point → (6, 12)
(b) Give the coordinates of the turning point of y = f(2x).
Stretch parallel to x-axis, scale factor 1/2 — divide x by 2:
Turning point → (3, 4)
Common Mistakes Students Make
- Getting the direction of f(x + a) wrong. Students see "+ 3" and move the graph right. It is the opposite — f(x + 3) moves left. Always ask yourself: "What value of x makes the inside equal the original x?" If f(x + 3), you need x + 3 = 0, so x = −3 — the graph shifts to where x = −3 was the zero, i.e. left.
- Confusing which coordinate changes. Changes inside the bracket affect x; changes outside affect y. Writing this on your working can save marks.
- Applying two transformations in the wrong order. If you are asked for f(2x + 1), this is f(2(x + ½)) — a horizontal stretch first (÷2) then a translation (left ½). Order matters.
- Forgetting to label the transformed graph. When sketching, always label the new graph with its equation and mark key points. Marks are often awarded for labelling, not just the shape.
- Thinking the y-intercept stays fixed for horizontal translations. It does not — when you shift left or right, the y-intercept changes too. Substitute x = 0 into the new function to find it.
Exam Tips for Graph Transformations
- Mark specific points first, then draw the curve through them. Pick the turning points, intercepts, and any named points given in the question. Map each one under the transformation, then join them up. Examiners award marks for correct points even if the sketch is rough.
- Write the transformation rule at the top of your working — e.g. "outside the bracket, so y-coordinates × 2." This keeps you on track and shows the examiner your reasoning.
- AQA often gives you a table of values or coordinates — make sure you transform every single point listed, not just two or three. Missing a point can cost you marks.
- On Edexcel, questions often ask for the equation of the transformed graph as well as the sketch. Be ready to write y = f(x + 3) or y = 2f(x) in the correct form.
- Check asymptotes. If the original graph has a vertical asymptote (e.g. y = 1/x has one at x = 0), a horizontal translation moves that asymptote too. Update it.
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