Model direct and inverse variation with powers and roots
Find a proportionality constant from given data · Use direct, inverse, power and root models · Check the domain and whether the relationship is plausible
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Formula rearrangement
- Powers and square roots
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
- Find a proportionality constant from given data
- Use direct, inverse, power and root models
- Check the domain and whether the relationship is plausible
Translate the stated relationship
Direct variation y ∝ x means y = kx; y ∝ x² means y = kx²; y ∝ √x means y = k√x on x ≥ 0. Inverse variation y ∝ 1/x means y = k/x with x ≠ 0; similarly k/x² or k/√x need the denominator non-zero and the root real.
Find k before predicting
Substitute the given pair to calculate k, write the full equation, then predict another value. In inverse-square variation, doubling the input quarters the output, not halves it. In inverse-root variation, quadrupling a positive input halves the output.
The model is a stated assumption, not something proved by one data pair. Check units, positivity and any context domain before extending a prediction.
Sources for this lesson
Pause and explain
In y = k/x² with positive x, what happens when x doubles?
Worked example
y varies inversely as x². If y = 12 at x = 2, find y at x = 4.
Show the worked solution
- 12 = k/4 gives k = 48.
- At 4, y = 48/16 = 3.
Answer y = 48/x²; predicted value 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.
y varies directly as x²; y = 18 when x = 3. Find the rule and y at x = 2.
Give me a hint
Substitute the known pair to find k.
Compare my reasoning
- 18 = 9k gives k = 2.
- At 2, y = 2×4 = 8.
y = 2x²; y = 8 at x = 2.
Look for these in your work
- Find k first
- Use the stated power
y varies inversely as x; y = 12 when x = 5. Find y at x = 8.
Give me a hint
Use yx = k.
Compare my reasoning
- k = 12×5 = 60.
- y = 60/8 = 7.5.
y = 60/x, x ≠ 0; predicted value 7.5.
Look for these in your work
- Retain the denominator restriction
- Check direction of change
A model gives y ∝ 1/√x for x > 0, with y = 10 at x = 4. Find y at x = 25 and explain why x = 0 is excluded.
Give me a hint
The denominator is a square root.
Compare my reasoning
- 10 = k/2 gives k = 20.
- At 25, y = 20/5 = 4. At zero the denominator would vanish.
y = 20/√x on x > 0; value 4 at x = 25; zero is excluded.
Look for these in your work
- Use the root correctly
- State the real non-zero domain
Common mistakes
- Inverse-square and inverse-linear changes differ.
- A proportionality sign is not a complete equation until k is found.
What can you explain now?
In y = k/x² with positive x, what happens when x doubles?
Compare with the explanation
y is divided by 4.
The denominator increases by a factor of 2².
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
In y = k/x² with positive x, what happens when x doubles?
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.