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Translating Graphs

A translation slides a graph without changing its shape. The trick everyone meets here: changes outside the function move it vertically (as expected), but changes inside the brackets move it horizontally the “wrong” way.

25 min Graphs and Transformations

Edexcel AS Level Maths: 4.5–4.7 Transformation of GraphsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Apply vertical translations f(x) + a
Apply horizontal translations f(x + a)
Understand why inside changes act in the opposite direction
Describe a translation as a vector

Vertical translations: f(x) + a

Adding a constant outside the function shifts the whole graph $\textbf{up}$ by $a$ (or down if $a$ is negative). This behaves exactly as you would expect.

y = f(x) + a \;\Rightarrow\; \text{shift up by } a

Outside the function ⟶ vertical movement, in the natural direction.

Horizontal translations: f(x + a)

Adding a constant $\textbf{inside}$ the brackets shifts the graph horizontally — but in the opposite direction to the sign. $f(x + a)$ moves $\textbf{left}$ by $a$ , while $f(x - a)$ moves $\textbf{right}$ by $a$ .

y = f(x + a) \;\Rightarrow\; \text{shift LEFT by } a

Inside the function ⟶ horizontal movement, the opposite way to the sign.

Example 1Describe the transformation from

y = x^2

y = (x - 3)^2

1The

-3

is inside the bracket, so it is a horizontal shift.

2Opposite to the sign:

-3

moves it right.

Answer

translation 3 units to the right

Tip — Inside the bracket = opposite direction. f(x − 3) moves RIGHT, not left.

Translations as vectors

A combined translation can be written as a column vector. Moving right by $a$ and up by $b$ is the vector $\begin{pmatrix} a \\ b \end{pmatrix}$ , taking $y = f(x)$ to $y = f(x - a) + b$ .

Formula recap

y = f(x) + a

Shift up by a (vertical).

y = f(x + a)

Shift LEFT by a (horizontal, opposite sign).

y = f(x - a) + b \;=\; \begin{pmatrix} a \\ b \end{pmatrix}

Combined translation vector.

Common mistakes to avoid

Moving f(x − 3) to the left.

Inside changes act oppositely: f(x − 3) moves the graph RIGHT.

Confusing inside and outside changes.

Outside the function → vertical; inside the brackets → horizontal.

Key takeaways

f(x) + a shifts up by a (vertical, as expected).
f(x + a) shifts LEFT by a; f(x − a) shifts right — opposite to the sign.
A translation can be written as a column vector.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$y = f(x) + 5$ translates the graph:
- Up 5
- Down 5
- Left 5
- Right 5
Show answer and explanation
Answer: Up 5
Outside the function shifts vertically, in the natural direction.
$y = f(x - 2)$ translates the graph:
- Right 2
- Left 2
- Up 2
- Down 2
Show answer and explanation
Answer: Right 2
Inside the bracket acts oppositely: −2 moves it right.
$y = f(x + 4)$ translates the graph:
- Left 4
- Right 4
- Up 4
- Down 4
Show answer and explanation
Answer: Left 4
+4 inside moves the graph left.
From $y = x^{2}$ to $y = (x + 1)^{2} - 3$ , the translation is:
- Left 1, down 3
- Right 1, down 3
- Left 1, up 3
- Right 1, up 3
Show answer and explanation
Answer: Left 1, down 3
+1 inside → left 1; −3 outside → down 3.
Which change moves a graph vertically?
- Adding outside the function
- Adding inside the brackets
- Multiplying x
- Negating x
Show answer and explanation
Answer: Adding outside the function
Outside changes affect the output (y), so move vertically.
The translation vector for $y = f(x - 3) + 2$ is:
- (3, 2)
- (−3, 2)
- (3, −2)
- (2, 3)
Show answer and explanation
Answer: (3, 2)
Right 3, up 2 → vector (3, 2).

Next step

Check your understanding

Work through questions on Translating Graphs. Use the feedback to decide what needs another look.

2 past-paper questions · 2 marks

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