Transform f(x) + k, kf(x) and f(x + k)
Distinguish vertical translation, vertical scaling and horizontal translation · Transform coordinates and domain restrictions · Explain the opposite sign of a horizontal shift
Section E3.12 · Extended · Translations and vertical scaling of graphs
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Function notation
- Graph coordinates
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Distinguish vertical translation, vertical scaling and horizontal translation
- Transform coordinates and domain restrictions
- Explain the opposite sign of a horizontal shift
Outside changes the output
If (p, q) lies on y = f(x), then (p, q + k) lies on f(x) + k: a vertical translation. For kf(x), the point is (p, kq): vertical scale factor |k|, with reflection in the x-axis when k < 0. At k = 0 the outputs collapse to zero on the original domain.
The domain stays the same for these two outside changes; an algebraically simplified zero expression does not automatically fill excluded original inputs.
Inside changes the input
For y = f(x + k), the old input p occurs when x + k = p, so the new point is (p − k, q). A positive k shifts the graph left, not right. The domain shifts too: if the original requires x ≥ 0, f(x + 2) requires x ≥ −2.
Apply one transformation at a time for combined expressions. This 0444 section lists f(x + k), not an obligation to replace it with f(kx); they are different transformations.
Sources for this lesson
Pause and explain
Which way does f(x + 2) shift the original graph?
Worked example
A point (2, 5) is on f. Where does it move for f(x) + 3, −2f(x) and f(x + 3)?
Show the worked solution
- Outside addition changes the height to 8.
- Outside scaling changes the height to −10.
- Inside addition requires x + 3 = 2, so the new x is −1.
Answer (2, 8), (2, −10), (−1, 5), respectively.
Try a topic test
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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.
A point (−1, 4) lies on f. Give its corresponding points on f(x) − 2 and 3f(x).
Give me a hint
Only change the output coordinate.
Compare my reasoning
- Subtracting 2 gives 4 − 2 = 2.
- Multiplying by 3 gives 12.
(−1, 2) and (−1, 12).
Look for these in your work
- Keep the input unchanged
- Apply the whole vertical change
For f(x) = x², find the vertex of f(x − 3) + 2 and expand its rule.
Give me a hint
Solve x − 3 = 0 for the shifted vertex input.
Compare my reasoning
- The vertex shifts from (0, 0) to (3, 2).
- (x − 3)² + 2 = x² − 6x + 11.
Vertex (3, 2); rule x² − 6x + 11.
Look for these in your work
- Use the inside sign correctly
- Check the expansion
For f(x) = 1/x, find the excluded input of f(x + 2), its output at x = 0, and the domain of 0f(x).
Give me a hint
Keep restrictions from the original evaluation of f.
Compare my reasoning
- f(x + 2) = 1/(x + 2), excluding x = −2; at 0 its value is 1/2.
- 0f(x) is zero where f is defined, so its domain still excludes 0.
Shifted exclusion −2; output 1/2 at 0. The domain of 0f(x) is x ≠ 0.
Look for these in your work
- Shift a domain break
- Do not erase an original undefined input
Common mistakes
- f(x + 3) shifts left by 3.
- A vertical multiplier changes all output heights, not input coordinates.
What can you explain now?
Which way does f(x + 2) shift the original graph?
Compare with the explanation
Left by 2 input units.
The same original input is reached two units earlier.
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Make it stick
Which way does f(x + 2) shift the original graph?
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.