Distinguish growth patterns and interpret rescaled time
Recognise constant first differences, second differences and ratios · Interpret the time interval in an exponential exponent · Explain eventual exponential growth without relying on one small table
Section Paper 4 · Historical Extended · 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
- Arithmetic, geometric and quadratic sequences
- Exponents
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
- Recognise constant first differences, second differences and ratios
- Interpret the time interval in an exponential exponent
- Explain eventual exponential growth without relying on one small table
Use equal input spacing
At equally spaced inputs, a non-constant linear model has constant first differences; a quadratic model has constant second differences; a non-zero exponential model has constant ratios. Unequal input spacing needs rates or powers that account for the spacing.
For P(t) = 80(1.01)¹²ᵗ with t in years, 1.01 is the multiplier per month: 12t counts months. One year's multiplier is 1.01¹², approximately 1.1268, not 1.12. For Q(t) = 50(1.2)ᵗ⁄¹⁰ with t in years, the 20% increase is per ten years, not per year.
Eventually versus immediately
A positive exponential with base greater than 1 eventually outgrows any fixed-degree polynomial as the input increases. For instance, 2ˣ is less than 100x² at x = 10 but greater at x = 20. The crossover depends on coefficients and the input domain.
For a positive polynomial at large consecutive integer inputs, its output ratio approaches 1; an exponential keeps its fixed ratio greater than 1, accumulating the advantage. This explains the long-term result; a few examples alone do not prove it. A base between zero and one models decay, so do not apply the growth statement to it.
Sources for this lesson
Pause and explain
In 50(1.2)ᵗ⁄¹⁰ with t in years, what period gives a factor of 1.2?
Worked example
Classify 3, 7, 11, 15; 2, 6, 12, 20; and 3, 6, 12, 24 at consecutive inputs.
Show the worked solution
- First list has constant first difference 4: linear.
- Second has first differences 4, 6, 8 and constant second difference 2: quadratic under that model assumption.
- Third has constant ratio 2: exponential under that model assumption.
Answer Linear, quadratic and exponential patterns, respectively.
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.
At inputs 0, 1, 2, 3 the outputs are 5, 9, 13, 17. Identify the linear rule assuming that pattern continues.
Give me a hint
Find the common difference and the output at zero.
Compare my reasoning
- The difference per input unit is 4.
- The intercept at input zero is 5.
f(x) = 4x + 5 under the linear-model assumption.
Look for these in your work
- Use equal spacing
- State the assumption
A balance is 200(1.05)⁴ᵗ, with t in years. What does 1.05 represent, and what is the one-year multiplier?
Give me a hint
The exponent counts quarters.
Compare my reasoning
- 4t is the number of quarters; 1.05 is 5% growth per quarter.
- One-year multiplier is 1.05⁴ = 1.21550625.
5% per quarter; annual multiplier 1.21550625, or about 21.55% annual growth.
Look for these in your work
- Distinguish period and annual rate
- Compound rather than add percentages
Compare 2ˣ with 100x² at x = 10 and 20. What do the checks establish, and what extra reasoning supports eventual dominance?
Give me a hint
Calculate both pairs; distinguish examples from a general argument.
Compare my reasoning
- At 10: 1024 < 10000. At 20: 1048576 > 40000.
- These check two inputs, not every future input. For large x the quadratic's successive ratio ((x + 1)/x)² tends to 1, while 2ˣ keeps ratio 2.
The exponential is smaller at 10 and larger at 20. Its fixed multiplier, compared with the polynomial's ratio tending to 1, explains eventual dominance.
Look for these in your work
- Do not overclaim from samples
- Require base greater than 1
Common mistakes
- Check equal input spacing before classifying differences or ratios.
- An exponent 12t or t/10 changes the time unit of the multiplier.
- Eventually outgrows does not mean larger at every input.
What can you explain now?
In 50(1.2)ᵗ⁄¹⁰ with t in years, what period gives a factor of 1.2?
Compare with the explanation
Ten years.
Increasing t by 10 increases the exponent by one.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
In 50(1.2)ᵗ⁄¹⁰ with t in years, what period gives a factor of 1.2?
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.