Exponential Function
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
An exponential function is a type of mathematical function in which the variable appears in the exponent. It models situations where the value of a quantity changes by a constant factor over equal intervals. The general form of an exponential function is:
Where:
a is the initial value or the starting point (the value of y when x = 0),
b is the base, a constant number (and b > 0, b does not equal 1),
x is the exponent (the independent variable).
Key Characteristics of Exponential Functions:
The function multiplies or divides at a constant rate. This means that each time the input x increases by 1, the output y is multiplied or divided by the same factor, b.
If b > 1, the function grows exponentially, meaning it multiplies by the same factor as x increases (exponential growth).
If 0 < b < 1, the function decays exponentially, meaning it is divided by the same factor as x increases (exponential decay).
The graph of an exponential function is a curve, not a straight line.
The rate of change is proportional to the current value of the function, causing rapid increases or decreases.