4.5 Translating Graphs
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Learning Objectives
- Understand how graphs shift horizontally and vertically
- Recognise the impact of adding or subtracting constants inside or outside a function
- Translate standard functions like quadratics, cubics, and exponentials
- Interpret translations algebraically and graphically
- Use notation such as f(x) → f(x ± a) or f(x) ± a
Key Concepts
What Is a Translation?
A **translation** moves a graph **without changing its shape** — it slides it up, down, left or right.
It’s one of the four basic transformations: translation, reflection, stretch, and compression.
Translations are often expressed in terms of changes to the function notation.
Vertical Translations: f(x) ± a
Adding or subtracting **outside** the function moves the graph **up or down**.
- f(x) + a ⇒ moves the graph **up** by a units
- f(x) − a ⇒ moves the graph **down** by a units
Example: y = x² + 3 is y = x² shifted **up** 3 units
Horizontal Translations: f(x ± a)
Adding or subtracting **inside** the function moves the graph **left or right**.
- f(x − a) ⇒ moves the graph **right** by a units
- f(x + a) ⇒ moves the graph **left** by a units
This feels 'reversed' but is correct because of how input values are affected.
Example: y = (x − 2)² is y = x² shifted **right** 2 units
Combined Translations
You can apply horizontal and vertical translations together.
Example: y = (x + 1)² − 4 is y = x² moved **left 1 and down 4**.
Always do horizontal (inside) and vertical (outside) separately.
General Notation: f(x) → f(x − a) + b
This is the standard format for describing translations.
It tells you to move the graph **right by a** and **up by b**.
Watch out: If signs are negative (e.g. f(x + a) − b), that means **left a**, **down b**.
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