Exponential Growth and Decay

Exponential growth and decay describe situations where the rate of change of a quantity is proportional to the current value. This means the amount increases or decreases by the same percentage in each time period.

• Growth: When a quantity increases over time (e.g. population, investments).
• Decay: When a quantity decreases over time (e.g. depreciation, radioactive decay).

General Formula

$\text{Final Amount} = \text{Initial Amount} \times (1 \pm r)^n $

• \( r \): rate of growth or decay as a decimal (e.g. $5\% → 0.05$)
• \( n \): number of time periods (years, months, etc.)
• Use \( +r \) for growth and \( -r \) for decay

Why Exponential?

The word “exponential” comes from the fact that the quantity is multiplied by a fixed number repeatedly — the variable is in the exponent. This leads to rapid changes compared to linear growth or decay.

Important Characteristics:

• Growth curves: become steeper over time.
• Decay curves: fall rapidly at first, then level off.
• Useful for modeling real-life processes like compound interest, inflation, population growth, depreciation, and cooling.

Graphical Interpretation:

• Exponential Growth: Curve slopes upwards — it gets steeper.
• Exponential Decay: Curve slopes downwards — it flattens out over time.