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. The general form of the equation is $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential decay is the decrease of a quantity at a rate proportional to its current value, leading to a rapid decline. The general form of the equation is $$y = a(1 - r)^t$$ , where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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 (decimal), 'n' is the number of times that interest is compounded per year, and 't' is the number of years.

To calculate the future value of an investment with exponential growth, use the formula $$A = P(1 + r)^t$$ , where 'A' is the future value, 'P' is the principal amount, 'r' is the growth rate, and 't' is the time in years.

Half-life is the time required for a quantity to reduce to half its initial value. In exponential decay, it can be calculated using the formula $$t_{1/2} = \frac{\text{ln}(2)}{r}$$ , where 'r' is the decay constant.

Linear functions have a constant rate of change and are represented by the equation $$y = mx + b$$ . Exponential functions have a variable rate of change that increases or decreases rapidly and are represented by $$y = ab^x$$ .

In an exponential function of the form $$y = ab^x$$ , the base 'b' determines the growth (if b > 1) or decay (if 0 < b < 1) rate of the function.

The initial value 'a' in an exponential function $$y = ab^x$$ represents the starting amount before any growth or decay occurs.

The growth factor is represented by $$1 + r$$ for growth and the decay factor by $$1 - r$$ for decay, where 'r' is the rate of growth or decay.

A real-world example of exponential growth is population growth, where the population increases rapidly as the number of individuals grows.