4.5PureCore

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 Video by Zeeshan Zamurred Graphs and Transformations

Edexcel AS Level Maths: 4.5–4.7 Transformation of GraphsWatch the full walkthrough before the notes below.

Open on YouTube

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 $up$ by $a$ (or down if $a$ is negative). This behaves exactly as you would expect.

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

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

Horizontal translations: f(x + a)

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

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

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

1The

- 3

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

2Opposite to the sign:

- 3

moves it right.

Answertranslation 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 $(a b)$ , 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 = (a b)

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

Ready to lock in Translating Graphs? Pick a mode and earn XP & Dobloons.