8-4 Transforming Exponential Expressions

Property

To write an exponential function from given information:
For growth: $y = a(1 + r)^t$ where $a$ is initial value and $r$ is growth rate as decimal
For decay: $y = a(1 - r)^t$ where $a$ is initial value and $r$ is decay rate as decimal

Examples

Property

Exponential functions can be rewritten using exponent properties to reveal different time periods or rates. The key property is $(a^m)^n = a^{mn}$, which allows us to transform $y = a(b)^t$ into equivalent forms like $y = a(b^{12})^{\frac{t}{12}}$ to show monthly rates from annual rates.

Examples

Property

The general exponential model $A(t) = a(1 + r)^t$ applies to any growth or decay scenario, not just finance. The same exponent-rewriting techniques used for compounding periods work here: