Exponential Growth & Decay Reference Cheat Sheet

Key Facts

• The standard exponential model is $y = a b^{x}$, where $a$ is the initial value, $b$ is the constant factor, and $x$ is the number of equal time intervals.
• Exponential growth has the form $y = a(1 + r)^{t}$, where $r$ is the growth rate written as a decimal.
• Exponential decay has the form $y = a(1 - r)^{t}$, where $r$ is the decay rate written as a decimal and $0 < r < 1$.
• The growth or decay factor can be found from two consecutive outputs using $b = \frac{y_{2}}{y_{1}}$ when the input increases by $1$.
• For compound interest, the model is $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $P$ is principal, $r$ is annual interest rate, $n$ is compounds per year, and $t$ is years.
• Continuous growth or decay uses $A = Pe^{rt}$, where $r > 0$ represents growth and $r < 0$ represents decay.
• Doubling time can be modeled by $A = P \cdot 2^{\frac{t}{d}}$, where $d$ is the time required for the quantity to double.
• Half-life can be modeled by $A = P\left(\frac{1}{2}\right)^{\frac{t}{h}}$, where $h$ is the time required for the quantity to decrease by half.

Vocabulary

Exponential Function

A function in which the variable appears in the exponent, commonly written as $y = a b^{x}$.

Initial Value

The starting amount in an exponential model, represented by $a$ in $y = a b^{x}$.

Growth Factor

The multiplier $b$ in an exponential growth model, where $b > 1$.

Decay Factor

The multiplier $b$ in an exponential decay model, where $0 < b < 1$.

Half-Life

The amount of time required for a decaying quantity to become half of its previous amount.

Compound Interest

Interest calculated on both the original principal and previously earned interest using $A = P\left(1 + \frac{r}{n}\right)^{nt}$.