Construct linear and exponential models from data
Construct linear or exponential functions from two points or a description · Account for unequal input spacing · Check model assumptions against additional data
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
- Linear gradients
- Exponent rules and 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
- Construct linear or exponential functions from two points or a description
- Account for unequal input spacing
- Check model assumptions against additional data
Fit the stated family
For a linear model y = mx + b through distinct inputs x₁ and x₂, m = (y₂ − y₁)/(x₂ − x₁); substitute one point to find b. Two points determine that line, but do not prove the actual process is linear.
For a positive exponential y = Arˣ with positive outputs and distinct inputs, y₂/y₁ = rˣ²⁻ˣ¹. Therefore r = (y₂/y₁)¹⁄⁽ˣ²⁻ˣ¹⁾, then A = y₁/rˣ¹. The ratio between observations is not necessarily the per-unit multiplier.
Words, sequences and validation
A fixed increase per unit input gives a linear model; a fixed percentage increase gives an exponential one. Arithmetic sequences are sampled linear rules; geometric sequences are sampled exponential rules, with indexing adjusted to the first supplied term.
State the intended input domain and units. Use additional observations to check consistency and consider whether extrapolation is plausible. Two points allow many families unless the model family is specified.
Sources for this lesson
Pause and explain
If an exponential doubles over two input units, is its per-unit multiplier 2?
Worked example
A positive exponential passes through (1, 6) and (3, 24). Find its rule.
Show the worked solution
- 24/6 = 4 = r², so the positive multiplier is r = 2.
- 6 = A×2 gives A = 3.
- Check 3×2³ = 24.
Answer y = 3×2ˣ.
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.
A linear function passes through (2, 7) and (5, 16). Find its equation.
Give me a hint
Find slope first, then intercept.
Compare my reasoning
- m = (16 − 7)/(5 − 2) = 3.
- 7 = 3×2 + b gives b = 1.
y = 3x + 1.
Look for these in your work
- Use distinct input spacing
- Check both points
A positive exponential passes through (0, 5) and (2, 45). Find its rule and its value at 1.
Give me a hint
The factor 9 occurs over two units.
Compare my reasoning
- A = 5; r² = 45/5 = 9, so r = 3.
- y(1) = 15.
y = 5×3ˣ; y(1) = 15.
Look for these in your work
- Use the positive multiplier
- Take the spacing-adjusted root
A lab population is 80 at hour 1 and 320 at hour 3. Assuming exponential growth, predict hour 2 and hour 4. Does this prove the model remains valid indefinitely?
Give me a hint
Find the per-hour multiplier from the two-hour ratio.
Compare my reasoning
- r² = 4, so r = 2. The model is P(t) = 40×2ᵗ.
- P(2) = 160 and P(4) = 640.
- Limited nutrients or changing conditions can invalidate long extrapolation.
160 at hour 2; 640 at hour 4 under the model. The two observations do not prove indefinite exponential growth.
Look for these in your work
- State the assumption
- Explain extrapolation limits
Common mistakes
- Do not use a two-step ratio as a one-step multiplier.
- Two observations alone do not identify the family without an assumption.
What can you explain now?
If an exponential doubles over two input units, is its per-unit multiplier 2?
Compare with the explanation
No; it is √2 for the positive exponential model.
The two-step multiplier is r².
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Make it stick
If an exponential doubles over two input units, is its per-unit multiplier 2?
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.