Graph Transformations
This topic covers 5 learning steps, guiding your child from the basics through to confident problem-solving. Each step includes a worked example and adaptive practice questions.
What Your Child Will Learn
- Translate graphs
f(x) + a shifts up, f(x - a) shifts right - Reflect graphs
-f(x) reflects in x-axis, f(-x) in y-axis - Stretch graphs
af(x) stretches vertically, f(ax) horizontally - Combined transformations
Apply multiple transformations to a graph - Describe transformations
Describe the transformation from one graph to another
Before This Topic
Your child should be comfortable with:
- Plotting Linear Graphs (Year 8)
- Translations (Year 7)
Common Mistakes
- Getting horizontal translations backwards — thinking f(x + 2) moves the graph 2 to the RIGHT instead of 2 to the LEFT
Inside the bracket, everything is "backwards" from what you might expect. f(x + 2) moves the graph 2 to the LEFT, and f(x - 2) moves it 2 to the RIGHT. Think of it this way: f(x + 2) = 0 when x = -2, so the graph shifts left. - Confusing vertical and horizontal stretches — thinking f(2x) stretches the graph by 2 when it actually squashes it by half
f(2x) is a horizontal stretch by a factor of 1/2 (it squashes towards the y-axis). Meanwhile, 2f(x) is a vertical stretch by a factor of 2 (it stretches away from the x-axis). Changes INSIDE the bracket affect x and are "opposite"; changes OUTSIDE affect y and are "as expected."
Tips for Parents
- Start with a simple graph like y = x² and show how each transformation changes it: y = x² + 3 moves it up 3, y = (x - 2)² moves it 2 to the right. Plot both on the same axes.
- Teach the key rule: changes OUTSIDE the function (e.g. 2f(x), f(x) + 3, -f(x)) affect the y-values directly. Changes INSIDE (e.g. f(2x), f(x - 3), f(-x)) affect the x-values and are "opposite" to what you expect.
- Use a graphing tool (like Desmos, which is free online) to let your child experiment: type y = x², then try y = (x - 3)², y = x² + 4, y = 2x², and see what happens in real time.
- Make a transformation checklist: translation (shift), reflection (flip), stretch (scale). For each one, ask: is the change inside or outside the bracket? Then apply the rule.
Key Words
- Translation — Shifting a graph up, down, left, or right without changing its shape — f(x) + 3 shifts up 3, f(x - 2) shifts right 2.
- Reflection — Flipping a graph — -f(x) reflects in the x-axis, f(-x) reflects in the y-axis.
- Stretch — Scaling a graph — 2f(x) stretches vertically by factor 2, f(2x) compresses horizontally by factor 1/2.
- f(x) notation — A way of writing functions — f(x) = x² means "the function of x is x squared." The f(x) tells you the y-value for any x.
- Transformation — Any change applied to a graph — translations, reflections, and stretches are all transformations.
Where This Fits
Before this topic: Children should be confident plotting linear and quadratic graphs, understand coordinates in all four quadrants, and know what translations mean geometrically.
After this topic: Graph transformations lead to sketching complex functions, trigonometric graph transformations, and iteration methods at GCSE and A-level.
How MathCraft Teaches This
In MathCraft, Graph Transformations is taught through the Coordinates & Statistics adventure track. Your child follows guided lessons with friendly characters, works through examples step by step, then practises with questions that adapt to their level.
The adaptive engine tracks mastery across all 5 steps, revisiting concepts your child finds tricky and advancing when they're ready. Parents can see detailed progress in the Parent Dashboard.
Practise Graph Transformations with MathCraft
Step-by-step lessons, worked examples, and adaptive practice — all wrapped in an adventure game your child will love.
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