Basic Exponential Growth and Decay

Exponential growth occurs when a quantity increases by a consistent percentage over a period of time, resulting in a rapid increase. For example, if a population doubles every year, it is experiencing exponential growth.

Exponential decay occurs when a quantity decreases by a consistent percentage over a period of time, leading to a rapid decrease. An example is radioactive decay, where a substance loses half of its mass in a fixed time period.

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.

In an exponential function y = a * b^x, the base 'b' represents the growth (b > 1) or decay (0 < b < 1) factor.

An exponential function graph will show a curve that either rises steeply (growth) or falls steeply (decay) rather than a straight line.

Linear functions increase or decrease by a constant amount, while exponential functions increase or decrease by a constant percentage.

An example is compound interest in finance, where the amount of money grows exponentially over time due to interest being calculated on the accumulated amount.

The y-intercept of an exponential function y = a * b^x is the value of 'a', which represents the initial amount when x = 0.

As x increases in an exponential decay function, the value of y approaches zero but never actually reaches it, creating a horizontal asymptote at y = 0.

The formula for exponential decay can be expressed as y = a * e^(-kt), where 'a' is the initial amount, 'k' is the decay constant, and 't' is time.