Exponential Decay and Growth

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to growth that accelerates over time. It can be modeled by the equation: $$y = a(1 + r)^t$$ where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be modeled by the equation: $$y = a(1 - r)^t$$ where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

In the equation, 'r' represents the decay rate as a decimal. For example, a decay rate of 5% would be represented as r = 0.05.

To calculate the remaining amount after exponential decay, use the formula: $$y = a(1 - r)^t$$ where 'a' is the initial amount, 'r' is the decay rate, and 't' is the time in years.

The growth model for a population that quadruples every day can be expressed as: $$y = 10(4)^x$$ where '10' is the initial population and 'x' is the number of days.

The formula for compound interest is: $$A = P(1 + r/n)^{nt}$$ where 'A' is the amount of money accumulated after n years, including interest, 'P' is the principal amount, 'r' is the annual interest rate, 'n' is the number of times that interest is compounded per year, and 't' is the number of years.

To find the population after a certain number of years with exponential growth, use the formula: $$P = P_0(1 + r)^t$$ where 'P_0' is the initial population, 'r' is the growth rate, and 't' is the number of years.

The initial value in exponential functions is the starting amount before any growth or decay occurs, often denoted as 'a' in the equations.

The time variable 't' represents the duration over which growth or decay occurs, affecting the final amount in exponential equations.

To determine the amount remaining after a certain time with a decay rate, apply the formula: $$y = a(1 - r)^t$$ where 'a' is the initial amount, 'r' is the decay rate, and 't' is the time.