Translate function graphs
Describe vertical and horizontal translations and write a transformed equation, accounting for the reversed sign inside f(x + k).
Section 3 · Functions
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does y = f(x) − 3 do?
What you will learn
- Recognise y = f(x) + k as a vertical translation
- Recognise y = f(x + k) as a horizontal translation
- Relate point coordinates to translated functions
Outside changes move outputs
y = f(x) + k adds k to every output, so positive k shifts the graph up and negative k shifts it down.
A point (x, y) on y = f(x) becomes (x, y + k). Shape and x-coordinates stay the same.
Inside changes move inputs in the opposite direction
y = f(x + k) shifts the graph k units left when k is positive. A point (x, y) maps to (x − k, y).
Thus f(x − 2) shifts right by 2. The syllabus considers integer k; check the sign inside the function carefully.
Pause and explain
What is the horizontal shift for y = f(x + 4)?
Worked example
Let f(x) = x². Write the function after translating the graph 2 right and 3 up.
Show the worked solution
- A shift right by 2 uses f(x − 2).
- A shift up by 3 adds 3 outside.
- So g(x) = f(x − 2) + 3 = (x − 2)² + 3.
Answer g(x) = (x − 2)² + 3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For f(x) = x², write the rule shifted 1 left and 4 down; state its vertex.
Give me a hint
Use f(x + 1) for the left shift and subtract 4 outside.
Compare my reasoning
- g(x) = f(x + 1) − 4.
- So g(x) = (x + 1)² − 4.
- The vertex moves from (0, 0) to (−1, −4).
g(x) = (x + 1)² − 4; vertex (−1, −4).
Look for these in your work
- Inside sign gives left shift
- Outside sign gives downward shift
- Vertex coordinates updated
A point (3, −2) lies on y = f(x). Find its image under y = f(x − 5) + 1.
Give me a hint
Translate right by 5 and up by 1.
Compare my reasoning
- The horizontal shift changes x = 3 to x = 8.
- The vertical shift changes y = −2 to y = −1.
- The image is (8, −1).
(8, −1).
Look for these in your work
- Horizontal direction correct
- Vertical change correct
- Coordinate order retained
A distance-time graph is transformed to y = f(x + 2) − 10. Describe the shift and map (6, 30).
Give me a hint
Translate left in x, then down in y.
Compare my reasoning
- The graph moves 2 units left and 10 down.
- The point's x-coordinate changes from 6 to 4.
- Its y-coordinate changes from 30 to 20, giving (4, 20).
Left 2 and down 10; (6, 30) maps to (4, 20).
Look for these in your work
- Both translations identified
- Coordinate changes follow the shifts
- Units can be attached to the axes
Common mistakes
- Reading positive k inside f(x + k) as a shift right
- Changing the graph shape during a translation
- Applying a horizontal shift to y-coordinates
- Mixing inside and outside changes
What can you explain now?
Where does (1, 4) move under y = f(x) + 2?
Compare with the explanation
(1, 6)
The x-coordinate stays fixed and the output increases by 2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why does adding inside f(x + k) shift left while adding outside shifts vertically?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.