Exponential Growth and Decay

Exponential Growth occurs when a quantity increases by a consistent percentage over a period of time, represented by the formula: $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential Decay occurs when a quantity decreases by a consistent percentage over time, represented by the formula: $$y = a(1 - r)^t$$ , where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

The general form of an exponential function is $$y = a(b)^x$$ , where 'a' is a constant, 'b' is the base (b > 0), and 'x' is the exponent.

An exponential function will show a constant ratio between consecutive outputs (y-values) when the inputs (x-values) are equally spaced.

Linear growth increases by a constant amount, while exponential growth increases by a constant percentage, leading to much larger values over time.

The base 'b' in an exponential function determines the growth or decay rate. If b > 1, it represents growth; if 0 < b < 1, it represents decay.

'r' represents the growth rate in the exponential growth formula $$y = a(1 + r)^t$$ , expressed as a decimal.

'r' represents the decay rate in the exponential decay formula $$y = a(1 - r)^t$$ , expressed as a decimal.

To convert a percentage growth rate to a decimal, divide the percentage by 100. For example, a 5% growth rate becomes 0.05.

The formula for continuous exponential growth is $$y = ae^{rt}$$ , where 'e' is Euler's number (approximately 2.718), 'a' is the initial amount, 'r' is the growth rate, and 't' is time.