A-Level — mathsStandard2 min read

Graph Transformations

Apply translations, stretches, and reflections to function graphs at A-Level.

Tutor AI Team2026-02-25T01:54:11.662Z

Graph transformations modify the graph of $y = f(x)$ systematically. Understanding these transformations is key to sketching curves at A-Level.

Transformations Summary

Transformation	Effect
$f(x) + a$	Translate up by $a$
$f(x) - a$	Translate down by $a$
$f(x + a)$	Translate left by $a$
$f(x - a)$	Translate right by $a$
$af(x)$	Vertical stretch, factor $a$
$f(ax)$	Horizontal stretch, factor $\frac{1}{a}$
$-f(x)$	Reflect in x-axis
$f(-x)$	Reflect in y-axis

Changes outside $f$ : affect $y$ (vertical, as expected). Changes inside $f$ : affect $x$ (horizontal, opposite direction).

Apply in order: inside first (horizontal), then outside (vertical).

$2f(x-3) + 1$ : translate right 3, then stretch vertically by 2, then translate up 1.

Problem

$y = x^2$ → $y = (x-2)^2 + 3$ : translate right 2, up 3. Vertex moves from $(0,0)$ to $(2,3)$ .

Solution

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Outside $f$ : vertical changes (as expected).
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Inside $f$ : horizontal changes (opposite direction).
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$f(x+a)$ : left by $a$ . $f(x-a)$ : right by $a$ .
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$af(x)$ : stretch vertically by $a$ . $f(ax)$ : compress horizontally by $a$ .

Tags:

#a-level#maths#pure#functions#transformations#graphs

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