Exponential Growth and Decay

This lesson covers how exponential functions are used to model real-world growth and decay processes. The Edexcel 9MA0 specification requires you to set up, use, and interpret exponential models, and to understand their limitations.

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The General Model

The standard exponential growth/decay model is:

y = Aeᵏᵗ

where:

• A = the initial value (when t = 0)
• k = the growth/decay constant
• t = time
• y = the value at time t

If k > 0, the model shows exponential growth. If k < 0, the model shows exponential decay.

When t = 0: y = Ae⁰ = A, confirming that A is the initial value.

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Exponential Growth Examples

Population Growth

A bacteria colony initially has 500 cells and doubles every 3 hours.

Model: N = 500 × 2^(t/3), or equivalently N = 500e^(kt) where k = ln(2)/3 ≈ 0.231

t (hours)	N (bacteria)
0	500
3	1000
6	2000
9	4000
12	8000

Compound Interest

An investment of £5000 grows at 4% per annum compounded continuously.