Exponential Functions

An exponential function has a variable in the exponent and models quantities that change by a constant factor over equal intervals.

For positive exponent variables, a base greater than one models exponential growth, while a base between zero and one models exponential decay.

A common example of exponential decay involves a radioactive tracer in medical imaging. Suppose a patient receives an initial dose of 800 milligrams with a six-hour half-life, meaning the tracer is halved every six hours.

Using time t in hours, the remaining tracer is modeled using an exponential function. Here, t over six gives the number of six-hour intervals, and the base one-half shows that the amount is halved in each interval. Because the base is between zero and one, this represents exponential decay.

The graph of this function starts at a y-intercept of 800. As t increases, the curve approaches zero, forming a horizontal asymptote at y equals zero.

For this model, the domain includes non-negative time values because time starts when the tracer is given.

The range includes tracer amounts greater than zero and up to 800, as the amount continually decreases toward zero.