GCSE

Graphs & Transformations

Moving and reflecting graphs — and why "inside the bracket" does the opposite of what you'd expect.

What Your Child Is Learning

Your child is learning to transform graphs — that means moving them up, down, left, or right (translations) and flipping them (reflections). They need to describe these transformations precisely using mathematical language and sketch what the transformed graph looks like.

The notation uses \(f(x)\) to represent any function. If you know what \(y = f(x)\) looks like, you can work out what \(y = f(x) + 3\) or \(y = f(x - 2)\) looks like without plotting every point.

How Schools Teach It Now

Students learn four core transformations using function notation:

Notation	What It Does	Example
\(f(x) + a\)	Translate up by \(a\)	\(f(x) + 3\) moves every point 3 units up
\(f(x + a)\)	Translate left by \(a\)	\(f(x + 2)\) moves every point 2 units left
\(-f(x)\)	Reflect in the x-axis	Every y-coordinate flips sign
\(f(-x)\)	Reflect in the y-axis	Every x-coordinate flips sign

The critical insight: changes inside the bracket (affecting \(x\)) do the opposite of what you'd expect. \(f(x + 2)\) moves the graph left, not right. Changes outside the bracket (affecting \(y\)) do exactly what you'd expect.

Schools insist on precise language: say "translate" not "move", and give the direction and distance. For example: "a translation of \(\begin{pmatrix} -2 \\ 0 \end{pmatrix}\)" or "a translation 2 units in the negative x-direction".

Common Mistakes to Watch For

\(f(x+2)\) moves the graph RIGHT: This is the single most common error. The \(+2\) is inside the bracket, so it moves left. Think of it this way: \(f(x+2)\) reaches each y-value 2 units sooner (earlier on the x-axis).
Confusing vertical and horizontal transformations: \(f(x) + 2\) (outside) moves up. \(f(x + 2)\) (inside) moves left. The bracket position determines the direction.
Using casual language instead of mathematical terms: examiners usually want "translate" and "reflect" — not "move" or "flip". Marks are often lost for imprecise descriptions.

Try It Together

Worked Example

Question: If \(y = x^2\), what does \(y = (x - 3)^2\) look like?

Step 1: The \(-3\) is inside the bracket (it's replacing \(x\) with \(x - 3\)).

Step 2: Inside the bracket = opposite direction. The minus means the graph moves right.

Step 3: By how much? By 3 units.

Answer: It's the same parabola (\(U\)-shape), shifted 3 units to the right. The vertex moves from \((0, 0)\) to \((3, 0)\).

A good follow-up question: "What about \(y = x^2 + 3\)?" That one's outside the bracket, so it moves straight up by 3. Vertex goes from \((0, 0)\) to \((0, 3)\).

Let Them Practise

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