Final answer:
An exponential function is expressed as a constant base raised to a variable exponent, illustrating rapid growth or decay. For example, bacterial growth can be modeled by an exponential function where the population doubles every hour, showing the accelerating rate of change that characterizes this growth.
Step-by-step explanation:
The exponential function is a mathematical expression in which a constant base is raised to a variable exponent. This type of function portrays rapid growth or decay and its rate of change increases or decreases multiplicatively as the function progresses. For instance, if we take the function f(x) = 2^x, this shows exponential growth with a base of 2. As x increases, the value of f(x) doubles each time, illustrating the accelerating nature of this growth.
In a real-world context, bacteria growth can be represented by an exponential function. If we start with 1000 bacteria and they double every hour, we can describe the population at hour 't' with P(t) = 1000 * 2^t. After one hour, P(1) = 2000, after two hours, P(2) = 4000, and after three hours, P(3) = 8000.
The rate of change in an exponential function reflects how the quantity increases by a particular factor with each step. For the bacteria example, the rate of change per hour is a factor of 2, which is the base of the exponential expression. This demonstrates the concept of exponential growth, where each generation of bacteria adds more to the overall population than the generation before.