Is y = f(2x) a stretch or a squash?

A squash. The horizontal scale factor is the reciprocal of 2, i.e. 12, so the graph is compressed towards the y-axis. A factor of f( 12 x) would stretch it out.

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B9PureCore

Graphs and Transformations

Q: Why does f(x + a) move the graph left instead of right?

Because the change happens to x before f acts. To reach the output the graph had at some x, the input must now be a smaller, so every feature slides in the negative (left) direction. Inside-the-bracket changes always reverse.

Q: How do I tell a vertical from a horizontal transformation?

Look at where the change is. Outside the function, af(x) or f(x)+a, it is vertical and behaves intuitively. Inside the function, f(ax) or f(x+a), it is horizontal and reversed/reciprocal.

Q: What is the difference between -f(x) and f(-x)?

-f(x) negates the output, reflecting the curve in the x-axis (flips it upside down). f(-x) negates the input, reflecting it in the y-axis (flips it left–right).

Once you know the shape of one graph, you can predict a whole family of related graphs by shifting, stretching or reflecting it. The one rule that trips everyone up: changes inside the bracket act horizontally and behave “backwards”.

30 min Video by TLMaths Algebra and Functions

The big picture

Transformations let you understand hundreds of graphs by learning a handful of parent shapes and how they move. It is enormously efficient — sketch $y = x^{2}$ once and you can instantly draw $y = (x - 3)^{2} + 2$ without plotting a single point. The genuine difficulty, and the thing worth really understanding, is the split between $outside$ the function (vertical, and intuitive) and $inside$ the function (horizontal, and counter-intuitive). Master that distinction and graph sketching, trigonometric graphs and curve-sketching in calculus all become far easier.

A-Level Maths B9-04: Graph Transformations — Translations OverviewWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Apply vertical and horizontal translations to a graph
Apply vertical and horizontal stretches, using the correct scale factor
Reflect a graph in the $x$ - or $y$ -axis
Describe and combine transformations, and track key points

Outside vs inside — the master rule

Every transformation is either applied to the whole function ( $outside$ the bracket) or to the input ( $inside$ the bracket). Outside changes are $vertical$ and do exactly what you expect. Inside changes are $horizontal$ and do the $opposite$ of what you expect.

Why the opposite? Because inside changes act on $x$ before the function runs. To get the same output as before, $x$ has to compensate in the reverse direction — so $f (x + 3)$ shifts $left$ , and $f (2 x)$ $squashes$ horizontally rather than stretching.

Why this works

Say it as a mantra: “outside = vertical = as expected; inside = horizontal = backwards.” Almost every transformation error is a violation of that one sentence.

Translations (shifts)

Adding a constant outside slides the graph vertically; adding inside slides it horizontally the “wrong” way.

y = f (x) + a \Rightarrow shift UP by a

Outside: vertical, intuitive.

y = f (x + a) \Rightarrow shift LEFT by a

Inside: horizontal, reversed.

f (x - a)

moves right.

1Inside the bracket:

x - 3

shifts the graph

right

by 3 (reversed).

2Outside:

+ 2

shifts it

up

by 2.

3So the vertex moves from

(0, 0)

(3, 2)

AnswerA translation right 3 and up 2 — vertex at

(3, 2)

Tip — A vector captures a translation neatly: $y = f (x - a) + b$ is a shift by $(a b)$ .

Stretches

Multiplying outside stretches vertically by that factor. Multiplying $x$ inside stretches horizontally by the $reciprocal$ — so $f (2 x)$ compresses by a factor of $\frac{1}{2}$ , not stretches by 2.

y = a f (x) \Rightarrow vertical stretch, factor a

Outside: multiply

y

-values by

a

y = f (a x) \Rightarrow horizontal stretch, factor \frac{1}{a}

Inside: reciprocal factor (a squash if

a > 1

1The change is inside, so it is horizontal.

2The scale factor is the reciprocal of 2, i.e.

\frac{1}{2}

3So the graph is squashed horizontally towards the

y

-axis by a factor of

\frac{1}{2}

AnswerHorizontal stretch of scale factor

\frac{1}{2}

(a compression).

Why this works

A vertical stretch multiplies every $y$ -coordinate; a horizontal stretch multiplies every $x$ -coordinate by the reciprocal. Tracking one or two key points through the transformation is the safest way to get it right.

Reflections and combining transformations

A minus sign outside flips the graph in the $x$ -axis; a minus inside flips it in the $y$ -axis. When several transformations combine, apply them in the right order — generally deal with inside-the-bracket changes together and outside changes together, and always check a couple of known points.

y = - f (x) \Rightarrow reflect in the x -axis

Outside minus:

y \to - y

y = f (- x) \Rightarrow reflect in the y -axis

Inside minus:

x \to - x

- f (x)

reflects in the

x

-axis:

y = 5 \to - 5

+ 1

shifts up 1:

- 5 \to - 4

3The

x

-coordinate is unchanged (both changes are outside/vertical).

AnswerThe point moves to

(2, - 4)

Tip — When asked to sketch, mark the images of key features — intercepts, turning points, asymptotes — rather than trying to move the whole curve at once.

Think like an examiner

State transformations precisely: “translation by $(30)$ ” or “horizontal stretch scale factor $\frac{1}{2}$ ”, not vague words like “moved across”.
Inside-the-bracket direction is the classic trap: $f (x + 2)$ is left, $f (x - 2)$ is right.
For a horizontal stretch $f (a x)$ , the scale factor is $\frac{1}{a}$ — writing $a$ loses the mark.
On a sketch, label the images of intercepts, turning points and asymptotes; an unlabelled curve rarely scores full marks.

Common misconceptions

Thinking

y = f (x + 3)

shifts the graph right.

Inside changes reverse:

f (x + 3)

shifts

left

by 3.

Saying

y = f (2 x)

is a horizontal stretch of factor 2.

It is a stretch of factor

\frac{1}{2}

(a squash) — the reciprocal.

Confusing

- f (x)

and

f (- x)

- f (x)

reflects in the

x

-axis;

f (- x)

reflects in the

y

-axis.

Applying combined transformations in a careless order.

Handle inside changes together and outside together, and verify with a known point.

Transformation summary

f (x) + a : up a

Vertical translation.

f (x + a) : left a

Horizontal translation (reversed).

a f (x) : vertical stretch a

Vertical stretch.

f (a x) : horizontal stretch \frac{1}{a}

Horizontal stretch (reciprocal).

- f (x), f (- x)

Reflections in

x

- and

y

-axis.

Stretch yourself

The graph of $y = f (x)$ has a maximum at $(2, 5)$ . State the coordinates of the maximum of $y = 3 f (x - 1)$ .

Hint — Deal with the inside change (horizontal) and the outside change (vertical) separately, and apply them to the point $(2, 5)$ .

Questions students ask

Why does

f (x + a)

move the graph left instead of right?

Because the change happens to

x

before

f

acts. To reach the output the graph had at some

x

, the input must now be

a

smaller, so every feature slides in the negative (left) direction. Inside-the-bracket changes always reverse.

How do I tell a vertical from a horizontal transformation?

Look at where the change is. Outside the function,

a f (x)

f (x) + a

, it is vertical and behaves intuitively. Inside the function,

f (a x)

f (x + a)

, it is horizontal and reversed/reciprocal.

y = f (2 x)

a stretch or a squash?

A squash. The horizontal scale factor is the reciprocal of 2, i.e.

\frac{1}{2}

, so the graph is compressed towards the

y

-axis. A factor of

f (\frac{1}{2} x)

would stretch it out.

What is the difference between

- f (x)

and

f (- x)

- f (x)

negates the output, reflecting the curve in the

x

-axis (flips it upside down).

f (- x)

negates the input, reflecting it in the

y

-axis (flips it left–right).

Key takeaways

Outside changes are vertical and intuitive; inside changes are horizontal and reversed.
$f (x) + a$ up; $f (x + a)$ left; $a f (x)$ vertical stretch $a$ ; $f (a x)$ horizontal stretch $\frac{1}{a}$ .
$- f (x)$ reflects in the $x$ -axis; $f (- x)$ reflects in the $y$ -axis.
Track key points (intercepts, turning points) through the transformation.
Describe transformations precisely, with direction and scale factor.

How this fits the course

Build on

The Modulus Function
Trigonometric graphs (later)soon

Leads to

Composite and Inverse Functions

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