IB — mathsStandard2 min read

Exponential Growth and Decay

Model real-world scenarios with exponential functions for IB Maths. Apply to population, radioactive decay, and finance.

Tutor AI Team2026-02-25T02:06:13.268Z

Exponential models describe quantities that grow or decay by a constant percentage. This is heavily tested in IB Math AI and AA.

Models

Growth: $y = a \cdot b^t$ where $b > 1$

Population doubling: $P = P_0 \cdot 2^{t/d}$ where $d$ = doubling time.

Decay: $y = a \cdot b^t$ where $0 < b < 1$

Half-life: $N = N_0 \cdot \left(\frac{1}{2}\right)^{t/h}$ where $h$ = half-life.

Continuous: $y = ae^{kt}$

$k > 0$ : growth. $k < 0$ : decay.

Worked Example: Population

Problem

Population 5000 grows 3% p.a. After 20 years: $5000 \times 1.03^{20} \approx 9031$ .

Solution

Worked Example: Half-life

Problem

Substance with half-life 8 hours. Starting with 200g, after 24 hours: $200 \times (\frac{1}{2})^3 = 25$ g.

Solution

Worked Example: Finding k

Problem

Bacteria: 100 at $t=0$ , 450 at $t=5$ . $450 = 100e^{5k}$ → $k = \frac{\ln 4.5}{5} \approx 0.301$ .

Solution

IB Exam Tips

Show the model equation clearly.
Use GDC for solving, but show algebraic setup.
Units and context matter for full marks.

Practice Problems

1. A car depreciates 15% per year from £25,000. Value after 6 years?
1. A substance has half-life 5 years. How long until 10% remains?

Want to check your answers and get step-by-step solutions?

Key Takeaways

✓
Growth: $b > 1$ or $k > 0$ . Decay: $0 < b < 1$ or $k < 0$ .
✓
Half-life: multiply by $\frac{1}{2}$ per period.
✓
Use logs to find time.

Tags:

#ib#maths#algebra#exponential#modelling

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Models

Growth: y=a⋅bty = a \cdot b^ty=a⋅bt where b>1b > 1b>1

Decay: y=a⋅bty = a \cdot b^ty=a⋅bt where 0<b<10 < b < 10<b<1

Continuous: y=aekty = ae^{kt}y=aekt

Worked Example: Population

Worked Example: Half-life

Worked Example: Finding k

IB Exam Tips

Practice Problems

Key Takeaways

Related Topics

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Ready to Ace Your IB maths?

Growth: $y = a \cdot b^t$ where $b > 1$

Decay: $y = a \cdot b^t$ where $0 < b < 1$

Continuous: $y = ae^{kt}$