Transformations of GraphsEdexcel IGCSE Maths: Revision notes

Section 1

What does f(x) + a do to a graph?

Adding a constant $a$ outside the function shifts the whole graph vertically.

$y = f(x) + a$ : translate by $\begin{pmatrix}0\a\end{pmatrix}$
If $a > 0$ , the graph moves up by $a$ units
If $a < 0$ , the graph moves down by $|a|$ units

Every $y$ -coordinate changes; every $x$ -coordinate stays the same. Roots (where the curve crosses the $x$ -axis) generally move, but turning points keep the same $x$ -value.

Key termstranslationvertical shift

Example

If (f(x) = x^2), then (f(x) + 3 = x^2 + 3) is the same parabola moved up 3 units. The minimum point moves from ((0,0)) to ((0,3)).

Common mistake

Do not confuse (f(x) + a) with (f(x + a)) — the position of (a) relative to the bracket completely changes whether the shift is vertical or horizontal.

Section 2

What does f(x + a) do to a graph?

Adding a constant inside the bracket shifts the graph horizontally — and in the opposite direction to what you might expect.

$y = f(x + a)$ : translate by $\begin{pmatrix}-a\\0\end{pmatrix}$
If $a > 0$ , the graph moves left by $a$ units
If $a < 0$ (i.e. $f(x - a)$ with $a>0$ ), the graph moves right by $a$ units

Every $x$ -coordinate changes; every $y$ -coordinate stays the same.

Key termshorizontal shift

Example

If (f(x) = x^2), then (f(x - 2) = (x-2)^2) moves the parabola right by 2, giving a minimum at ((2, 0)).

Exam tip

Trick to remember direction: set the bracket equal to zero. For (f(x+3)), the graph behaves like (f(x)) did at (x=0) when (x+3=0), i.e. (x=-3) — so it has shifted left.

Section 3

What do -f(x) and f(-x) do?

These are reflections, not translations.

(y = -f(x)): reflects the graph in the (x)-axis. Every (y)-value is negated; (x)-intercepts stay fixed.
(y = f(-x)): reflects the graph in the (y)-axis. Every (x)-value is negated; (y)-intercept stays fixed.

Both keep the same overall shape, just flipped.

Key termsreflection

Common mistake

Students often mix up which axis each reflects in. Remember: the minus sign attached to (x) (inside the bracket) reflects in the (y)-axis; the minus sign attached to the whole function (outside the bracket) reflects in the (x)-axis.

Think of it like this

Think of (-f(x)) as flipping the picture upside down (over a horizontal mirror on the x-axis), and (f(-x)) as flipping it left-to-right (over a vertical mirror on the y-axis).

Section 4

What does af(x) do (stretches)?

Multiplying the whole function by a constant $a$ produces a vertical stretch (or squash).

$y = af(x)$ : stretch vertically by scale factor $a$ , parallel to the $y$ -axis
If $|a| > 1$ , the graph is stretched further from the $x$ -axis
If $0 < |a| < 1$ , the graph is squashed towards the $x$ -axis
If $a$ is negative, it also includes a reflection in the $x$ -axis

$x$ -intercepts stay in the same place (since $a \times 0 = 0$ ), but all other $y$ -coordinates are multiplied by $a$ .

Key termsvertical stretchscale factor

Example

If $f(x) = \sin(x)$ has amplitude 1, then $y = 3f(x) = 3\sin(x)$ has amplitude 3 — the curve is stretched vertically by scale factor 3.

Exam tip

On IGCSE papers, af(x) almost always appears as a vertical stretch. Horizontal stretches (f(ax)) are rarely tested at this level, but if asked, f(ax) stretches horizontally by scale factor 1/a.

Section 5

How do I tackle combined transformations?

Exam questions often combine two transformations, e.g. (y = f(x-2) + 3) or (y = -f(x) + 1). Apply them one at a time, in the order they act on (x) and then on the output.

A reliable method:

Identify each transformation separately (shift, reflect, stretch).
Apply the transformation closest to (x) first (inside the bracket), then work outward.
Track what happens to a few key points (e.g. the vertex, roots, or a labelled coordinate given in the question) rather than the whole curve.

Exam questions frequently give you a sketch of (y=f(x)) with labelled points and ask you to state the new coordinates after a transformation — you do not need the algebraic equation of (f).

Key termscombined transformation

Example

Given a point ((2, 5)) on (y=f(x)), find the image point on (y = -f(x-1)). First shift right by 1: ((3,5)). Then reflect in the x-axis: ((3,-5)).

Common mistake

Do not apply the vertical shift before the horizontal shift is done correctly, or apply operations in the wrong order — always resolve what's happening inside the bracket to x first.

Must Know

$f(x) + a$ : vertical translation by $\begin{pmatrix}0\a\end{pmatrix}$
$f(x + a)$ : horizontal translation by $\begin{pmatrix}-a\\0\end{pmatrix}$ (opposite sign to what you'd expect)
$-f(x)$ : reflection in the $x$ -axis
$f(-x)$ : reflection in the $y$ -axis
$af(x)$ : vertical stretch, scale factor $a$ , from the $x$ -axis
For combined transformations, track key coordinates through each step in turn rather than trying to do it all at once

That's the notes covered.

Carry on to the next subtopic.