What is the standard form of an exponential function?

The standard form of an exponential function is represented as ( f(x) = ab^x ), where ( a ) is a constant that represents the initial value (or the y-intercept when ( x = 0 )) and ( b ) is the base of the exponential that determines the rate of growth or decay. In this form, ( b ) must be a positive real number not equal to 1.

This form captures the key characteristic of exponential functions, which is that the output grows (or decays) multiplicatively based on the value of ( x ). As ( x ) increases, the function's value increases rapidly if ( b > 1 ) (exponential growth) or decreases rapidly if ( 0 < b < 1 ) (exponential decay). The significance of ( a ) allows for vertical stretching or shrinking of the graph, depending on whether ( a ) is greater than or less than 1.

Understanding this form is essential in distinguishing exponential functions from other types of functions, such as quadratic functions (which have a different form involving squared terms), linear functions (typically in the form ( mx + b )), and rational functions. Each of these functions behaves