Transformations of Graphs and Functions — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers what happens to a graph when its equation is changed. By the end of this guide you should be able to identify and describe translations, reflections and stretches from function notation; work out where a given point moves to; and write the equation of a graph after a described transformation.

You should also be able to describe a transformation fully, in the form examiners require, and apply transformations to graphs you have not been given an equation for — including the sine and cosine curves.

The organising idea runs through every case: a change written outside f( ) does what it says; a change written inside f( ) does the opposite. Adding 3 outside moves the graph up 3. Adding 3 inside moves it left 3. Multiplying outside stretches the y-direction by that factor; multiplying inside stretches the x-direction by the reciprocal. If you remember nothing else, remember outside–as-expected, inside–opposite.

Key terms and definitions

Transformation — any change that moves or reshapes a graph.

Translation — a slide, with no turning and no change of size or shape. Described by a column vector.

Reflection — a flip in a mirror line, which at GCSE is always one of the axes.

Stretch — a change of size in one direction only, described by a scale factor and the direction it acts in.

Function notation — writing a graph as y = f(x) so that a change to the equation can be described without knowing what f actually is.

Invariant point — a point that does not move under the transformation. Points on the x-axis are invariant under a vertical stretch; points on the mirror line are invariant under a reflection.

Core concepts

Why "inside" behaves the opposite way

The rule feels arbitrary until you see where it comes from, and seeing it once makes the rule stick.

Take y = f(x − 2). To get the same output that f gave at x = 3, the new function needs x − 2 = 3, so x must be 5. Every output has therefore shifted to an x-value 2 larger, and the graph has moved 2 to the right — even though the equation contains a minus sign.

The change inside the bracket applies to the input before f acts, so the graph has to move the opposite way to compensate. Outside changes apply to the output after f has acted, so they behave exactly as written.

Vertical translations

y = f(x) + a moves the graph up by a. If a is negative it moves down. The translation vector is (0, a), written as a column.

Every y-coordinate increases by a and every x-coordinate is unchanged. So if (4, 9) lies on y = f(x), then (4, 11) lies on y = f(x) + 2.

A minimum at (1, −2) becomes a minimum at (1, 4) on y = f(x) + 6: the x-position of the turning point does not move.

Horizontal translations

y = f(x + a) moves the graph left by a, and y = f(x − a) moves it right. The translation vector is (−a, 0).

The clearest concrete case is the parabola. The graph of y = (x − 2)² is y = x² moved 2 to the right, with its vertex at (2, 0). The graph of y = (x + 5)² is moved 5 to the left, vertex at (−5, 0).

Here the y-coordinates are unchanged and every x-coordinate shifts. If (3, 7) lies on y = f(x), then on y = f(x − 1) the same height occurs one unit further right, at (4, 7).

Reflections

y = −f(x) reflects the graph in the x-axis. The minus is outside, applied to the output, so every y-coordinate changes sign: (2, 5) becomes (2, −5).

y = f(−x) reflects the graph in the y-axis. The minus is inside, applied to the input, so every x-coordinate changes sign: (−2, 3) becomes (2, 3).

The pair is easy to confuse, and the outside/inside rule sorts it out: outside changes affect y, so the flip is vertical, which is a reflection in the x-axis.

Stretches

y = af(x) is a vertical stretch, scale factor a. Every y-coordinate is multiplied by a. Points on the x-axis do not move, because multiplying zero by a leaves zero — they are the invariant points.

y = f(ax) is a horizontal stretch, scale factor 1/a. Every x-coordinate is divided by a. So y = f(2x) squashes the graph to half its width, and y = f(x/2) stretches it to twice its width. Points on the y-axis are invariant.

A scale factor between 0 and 1 is still called a stretch, even though the graph gets smaller in that direction. There is no separate word for it at GCSE.

Transformations of the sine and cosine curves

These are the graphs most often used to test the rules, because their key features are easy to state.

y = sin x oscillates between −1 and 1. So y = sin x + 1 oscillates between 0 and 2, with a maximum value of 2 — a vertical translation. And y = 3 sin x oscillates between −3 and 3, a vertical stretch of scale factor 3.

y = sin 2x completes two full cycles where y = sin x completes one, because the horizontal stretch has scale factor ½.

Describing a transformation fully

Marks are awarded for a complete description, and a partial one scores nothing even when the type is right.

For a translation, give the word "translation" and the vector. "Moved down 4" is not enough; write translation by (0, −4).

For a reflection, give the word "reflection" and the mirror line — "in the x-axis", not just "reflected".

For a stretch, give the word "stretch", the scale factor, and the direction — "stretch, scale factor 3, in the y-direction".

And describe one transformation, not two. If a question asks for a single transformation, saying "reflect then translate" suggests the single one was not identified.

Worked examples

Example 1: Identifying a translation from an equation

The graph of y = x² is transformed to y = (x + 3)² − 4. Describe the transformation and give the coordinates of the vertex.

The + 3 is inside the bracket, so it acts the opposite way: a shift of 3 to the left. The − 4 is outside, so it does what it says: a shift of 4 down.

Together this is a translation by the vector (−3, −4).

The vertex of y = x² is at (0, 0), so the new vertex is at (−3, −4).

Example 2: Tracking a point

The point (6, 2) lies on y = f(x). Find its image on y = 3f(x) and on y = f(3x).

For y = 3f(x) the 3 is outside, so it multiplies the output. The y-coordinate is tripled and the x-coordinate is unchanged: (6, 6).

For y = f(3x) the 3 is inside, so it acts on the input and the effect is the reciprocal. The x-coordinate is divided by 3 and the y-coordinate is unchanged: (2, 2).

The two answers differ completely, which is exactly why the position of the 3 has to be checked before anything is calculated.

Example 3: Writing the equation from a description

The graph of y = f(x) is reflected in the y-axis and the result is then stretched vertically by scale factor 2. Write the equation of the final graph.

A reflection in the y-axis changes the input's sign, giving y = f(−x).

A vertical stretch of scale factor 2 multiplies the whole output by 2, giving y = 2f(−x).

The order the operations are written in matters here only because one acts inside and one outside; writing y = 2f(x) after the reflection would have lost the reflection entirely.

Common mistakes and how to avoid them

Getting horizontal translations backwards. f(x + a) moves the graph left. Test it on y = (x − 2)², whose vertex is at x = 2, to settle the direction in a couple of seconds.

Confusing the two reflections. −f(x) is in the x-axis; f(−x) is in the y-axis. Outside changes affect y, so the flip is vertical.

Using a as the horizontal stretch factor in f(ax). The scale factor is 1/a: y = f(2x) squashes to half the width.

Changing both coordinates. A vertical transformation leaves x alone and a horizontal one leaves y alone. Only ever one of the two changes.

Giving an incomplete description. A translation needs its vector, a reflection needs its mirror line, and a stretch needs both a scale factor and a direction.

Moving the turning point the wrong way under a vertical translation. The x-coordinate of a maximum or minimum is unchanged by y = f(x) + a; only its height moves.

Describing two transformations when one was asked for. Re-read the question before writing the answer.

Exam technique for "Transformations of Graphs and Functions"

Look at the position of the change before anything else — inside the bracket or outside it. That single check decides the direction of every rule in this topic.

Write the vector, axis or scale factor down as part of the answer sentence, not as an afterthought. These are the parts that carry the marks.

When tracking a point, write the two coordinates separately and state which one is unchanged. It prevents the common error of adjusting both.

For sketch questions, mark the image of two or three key points — a turning point, an axis intercept — and draw the curve through them rather than attempting the whole shape freehand.

When the question gives a specific curve such as y = x², use its known features to check your answer. If you claim y = (x + 3)² is a shift to the right, substituting x = −3 gives y = 0, which shows the vertex is at −3, and the claim is wrong.

Quick revision summary

Outside f( ) behaves as written; inside f( ) behaves the opposite way. Every rule below is that one statement.

y = f(x) + a is a translation by (0, a) — up for positive a. The x-coordinates are unchanged.

y = f(x + a) is a translation by (−a, 0) — left for positive a. The y-coordinates are unchanged. So y = (x − 2)² has its vertex at (2, 0).

y = −f(x) is a reflection in the x-axis; y = f(−x) is a reflection in the y-axis.

y = af(x) is a vertical stretch of scale factor a, with the x-axis invariant. y = f(ax) is a horizontal stretch of scale factor 1/a, with the y-axis invariant.

Describe a transformation fully: translation plus vector, reflection plus mirror line, or stretch plus scale factor and direction.

Only one coordinate changes at a time — vertical transformations leave x alone, horizontal ones leave y alone.