What does the variable 'b' represent in the exponential function's rate of change?

In the context of an exponential function, the variable 'b' represents the growth or decay factor, which significantly influences how the function behaves over time. Specifically, in the function of the form (y = a \cdot b^x), 'a' typically indicates the initial value of the function when (x = 0), while 'b' indicates how the function changes as (x) increases.

If 'b' is greater than 1, the function exhibits exponential growth, meaning that the value of 'y' increases rapidly as 'x' becomes larger. Conversely, if 'b' is between 0 and 1, the function represents exponential decay, where 'y' decreases as 'x' increases. Thus, 'b' effectively determines the rate at which these changes occur, and it is this characteristic that qualifies it as the growth or decay factor in the exponential function.

Understanding the role of 'b' is crucial for interpreting exponential models, especially in real-world applications like population growth, radioactive decay, or compound interest. Recognizing that 'b' influences the steepness of the curve helps in analyzing how quickly or slowly the process represented by the function progresses over time.