GCSE Maths · Algebra

Graph transformations GCSE Questions and Worked Answers

A graph transformation moves every point according to one rule. Adding outside f changes the output (height); changing the input inside f changes where that output occurs. GCSE transformations here are translations and reflections.

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Start with the meaning

What you need to know about graph transformations

A point on a graph is like a position on a map: the first coordinate tells you how far across, and the second tells you how high. For example, (2, 5) means x = 2 and y = 5. Moving that point three units up gives (2, 8). Moving every point three units up moves the whole graph without changing its shape.

See the idea first

The output stays the same, but occurs later

A function is a rule that gives an output for an input. We write its output as f(x); y = f(x) draws the input/output pairs. For f(x) = x², input 0 gives output 0. In y = f(x − 2), the bracket reaches that same input 0 when x = 2. That is why the graph moves right, not left.

Green: y = x². Brown: y = (x − 2)². The same outputs occur 2 units further right. The vertices are marked.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Move a graph vertically

What the problem asks: The point (3, 7) lies on y = f(x). Where is its image on y = f(x) + 4?

How to solve it: The output 7 becomes 7 + 4 = 11 while the input stays 3, so the point becomes (3, 11). Every point moves 4 units up.

Move a graph horizontally

What the problem asks: Where does (3, 7) move on y = f(x + 2)?

How to solve it: The bracket must still feed 3 into f. Solve x + 2 = 3, so the new x is 1. The point becomes (1, 7): 2 units left.

Reflect a graph

What the problem asks: Where does (3, 7) move on y = −f(x), and on y = f(−x)?

How to solve it: −f(x) changes the output sign, giving (3, −7), a reflection in the x-axis. f(−x) needs input x = −3 to feed 3 into f, giving (−3, 7), a reflection in the y-axis.

A reliable routine

Transform a known point or a sketch

Use this for translations and reflections of a given function graph. Following how a known input/output pair changes explains the direction and avoids relying on a sign mnemonic.

Choose a labelled point, such as a vertex or intercept.
For a change inside f, find the new x that supplies the old input.
Apply any change outside f to the old output.
Repeat for key points and draw the same translated or reflected shape.

Check: A full description names the transformation and direction/distance or mirror axis. Stretch transformations are outside this guide.

Fully worked

Graph transformations GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Vertical translation

Higher only2 marks

Question

P = (−1, 3) lies on y = f(x). Find its image on $y=f(x)+5$ .

The input is unchanged.

y_{\text{new}}=3+5=8

The image is (−1, 8), a move 5 units up.

Example 2

Horizontal translation

Higher only3 marks

Question

Q = (4, −2) lies on y = f(x). Find its image on $y=f(x-3)$ .

To reproduce the old input 4, the bracket must equal 4.

x-3=4

x=7

The output remains −2, so Q moves to (7, −2).

Example 3

Reflection in the x-axis

Higher only2 marks

Question

The graph y = f(x) has a maximum at (2, 6). Describe the corresponding turning point on $y=-f(x)$ .

The output changes sign:

6\longmapsto-6

The point is (2, −6). A maximum becomes a minimum because all heights are reversed.

Example 4

Reflection in the y-axis

Higher only2 marks

Question

A root of y = f(x) is x = −5. Find the corresponding root of $y=f(-x)$ .

A root is a point with output zero. Set the new input equal to the old one:

-x=-5

x=5

The root is x = 5, the reflection of (−5, 0) in the y-axis.

Example 5

Two changes

Higher only3 marks

Question

The minimum of y = f(x) is (−2, 1). Find the minimum of $y=f(x+4)-3$ .

First reproduce the input −2:

x+4=-2

x=-6

Then change the output:

1-3=-2

The minimum is (−6, −2). Translation preserves whether a point is a minimum.

Example 6

Write the new equation

Higher only3 marks

Question

The graph $y=x^2$ is translated 3 units right and 2 units up. Write its equation and vertex.

Moving right means the old input is now x − 3. Moving up adds 2 to the output.

y=(x-3)^2+2

The vertex moves from (0, 0) to (3, 2). Check: at x = 3, the squared part is zero.

10 original questions · total 20 marks

Graph transformations GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Up

Higher only1 mark

(1, 2) lies on y = f(x). Find its image on $y=f(x)+7$ .

Show worked answer

2+7=9

The image is (1, 9).

Down

Higher only1 mark

(−3, 8) lies on y = f(x). Find its image on $y=f(x)-6$ .

Show worked answer

8-6=2

The image is (−3, 2).

Left

Higher only2 marks

(5, 4) lies on y = f(x). Find its image on $y=f(x+3)$ .

Show worked answer

x+3=5

x=2

The image is (2, 4).

Right

Higher only2 marks

(−2, 7) lies on y = f(x). Find its image on $y=f(x-5)$ .

Show worked answer

x-5=-2

x=3

The image is (3, 7).

Change the height sign

Higher only2 marks

Find the image of (−4, −3) on $y=-f(x)$ , given that it lies on y = f(x).

Show worked answer

The x-coordinate stays −4 and the output becomes −(−3) = 3. The image is (−4, 3), reflected in the x-axis.

Change the input sign

Higher only2 marks

Find the image of (−4, −3) on $y=f(-x)$ , given that it lies on y = f(x).

Show worked answer

-x=-4\implies x=4

The image is (4, −3), reflected in the y-axis.

Combined translation

Higher only3 marks

The vertex of y = f(x) is (2, −1). Find its image on $y=f(x-4)+6$ .

Show worked answer

x-4=2\implies x=6

-1+6=5

The vertex becomes (6, 5).

Describe precisely

Higher only2 marks

Describe the transformation from y = f(x) to $y=f(x+1)-4$ .

Show worked answer

Translation 1 unit left and 4 units down, with vector $\binom{-1}{-4}$ . Inside +1 means the old input is reached one unit further left; outside −4 lowers every output.

Reflect a parabola

Higher only2 marks

Reflect $y=x^2+3$ in the x-axis. Write the new equation.

Show worked answer

Negate the whole output:

y=-(x^2+3)

y=-x^2-3

Both terms change sign.

Find the missing shift

Higher only3 marks

A labelled point (1, 4) on y = f(x) becomes (6, 2) on $y=f(x-a)+b$ . Find a and b.

Show worked answer

The point moves 5 right and 2 down.

6-a=1\implies a=5

4+b=2\implies b=-2

Examiner-style feedback

Common graph transformations mistakes

Treating an inside sign like an outside sign

For f(x + 3), solve x + 3 = the old input. That puts the point 3 left.

Confusing mirror axes

Reflection in the x-axis changes y; reflection in the y-axis changes x.

Negating only one term

Reflecting y = x² + 3 in the x-axis gives −(x² + 3), not −x² + 3.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Use a known input/output pair.
Inside changes affect the input location.
Outside changes affect the output height.
Check the transformed key points.

Quick answers

Graph transformations FAQ

What does f(x) mean?

The output of a rule f at input x. It is not f multiplied by x.

Why does the sign seem reversed for horizontal moves?

You are finding the new input position that makes the bracket equal to the old input. For x − 3 to equal 2, x must be 5.

Build connected skills

What to revise next

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Recognise vertices and intercepts.

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Content standards

Curriculum and rights review

A13 Higher-tier content. All definitions and worked tasks are introduced from simple inputs. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content