Transforming graphs y = f(x) – GCSE Maths

Higher · Grade 8 · Non-calculator

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$y=f(x)+a$ is a vertical shift. $y=f(x+a)$ is a horizontal shift the opposite way (left if $a>0$). $y=-f(x)$ reflects in the $x$-axis. $y=f(-x)$ reflects in the $y$-axis. $y=af(x)$ stretches in $y$; $y=f(ax)$ stretches in $x$ by factor $\frac{1}{a}$. Apply one at a time if two transformations sit together, and watch the order.

A point $(2,5)$ on $y=f(x)$ becomes $(2,8)$ on $y=f(x)+3$ and $(-2,5)$ on $y=f(-x)$. Sketch the parent faintly, then the image boldly. Name the transformation in full: 'translation by $\left(\begin{array}{c}0\\ 3\end{array}\right)$', not just 'up'.

Outside the function moves $y$: $f(x)+a$ is up, $-f(x)$ is a reflection in $x$-axis. Inside moves $x$, and the $x$-shift is the opposite of the sign: $f(x+2)$ is 2 left. $f(-x)$ reflects in the $y$-axis. Apply one transformation at a time to a known point.

Objectives

• Move a graph left or right from inside the bracket.
• Reflect a graph with -f(x) or f(-x).
• Track a point, not only the whole curve.

Key points

• Recognise the four transformations from the algebra: $f(x)+a$, $f(x+a)$, $-f(x)$ and $f(-x)$.
• Move a single point (such as a turning point or intercept) under each transformation.
• Sketch the transformed graph on the same axes as the original, keeping its shape.
• Describe a transformation in words: 'translation by $\left(\genfrac{}{}{0ex}{}{0}{a}\right)$' or 'reflection in the $x$-axis'.

Worked examples

Questions and answers

GCSE Maths transforming graphs y = f(x) questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Saying f(x+2) moves the graph right

   A plus inside the bracket moves the graph left

2. Mixing up -f(x) and f(-x)

   A minus outside flips y, a reflection in the x-axis. A minus inside flips x, a reflection in the y-axis

3. Changing x for f(x)+a

   A number outside the bracket only changes the y-coordinate

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Before sketching, write the transformed coordinates of the turning point and the intercepts — those labelled points are where the marks are.
• For $f(x+a)$, a quick check: the new graph must equal the old one when the inside is the same, so the point where $x+a=0$ is the old $x=0$ point.

Printable worksheet

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More on Transforming graphs y = f(x)

• Transforming graphs y = f(x) worked examples and method
  The method in order, key facts and the full lesson explanation
• Transforming graphs y = f(x) practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.