The definition of exponential functions are discussed using graphs and values. The properties such as domain, range, horizontal asymptotes, x and y intercepts are also presented. The conditions under which an exponential function increases or decreases are also investigated.

## Definition of the Exponential Function
The basic exponential function is defined by
\[ f(x) = B^{x} \]
Example 1: Table of values and graphs of exponential functions with base greater than 1
Example 2: Table of values and graphs of exponential functions with base less than 1
## Properties of the exponential functions
From the above values and graphs we conclude the following properties
## Change any exponential functions \( B^x \) to a natural exponential of the form \(e^{kx}\)
Let \( f(x) = B^x \) be an exponential of base B.
## Explore the more general exponential functions using an app
Input boxes are used to change parameters included in the definition of the more general exponential function of the form
\[ f(x) = a B^{b(x+c)} + d \]
- Set a to 1, b to 1, c to 0, d to 0 and the base B to 2. Does the graph of function f increase or decrease?
- Set a to 1, b to 1, c to 0, d to 0 and the base B to 4. Does the graph of function f increase or decrease?
- Set a to 1, b to 1, c to 0, d to 0 and the base B to 0.4. Does the graph of function f increase or decrease?
- Set a to 1, b to 1, c to 0, d to 0 and the base B to 0.9. Does the graph of function f increase or decrease?
## Increase and Decrease of the Exponential Functions
Set a to 1, b to 1 , c to 0, d to 0 and change base B so that B greater than 1. Note that as long as B greater than 1, the exponential function B ## Range and Horizontal Asymptote of the Exponential Functions
Set a to 1, b to 1, c and d to zero. Set base B values greater than 1 and note the following: as x increases, B ## Shifting, Scaling and Reflection of the Exponential FunctionsWe now investigate the effects of parameters a, b, c and d on the properties of the graph of function f defined by:
\[ f(x) = a B^{b(x+c)} + d \]
Set B = e, b = 1, c = 0 and d = 0 and Explore the effects of parameter a (vertical scaling) on the graph of f.
You may want to work through a tutorial on finding Exponential Function Given its Graph. It is a tutorial that complements the one on this page.
## More References and LinksCalculate Exponentials and Logarithms to any Base.Graphing Exponential Functions natural logarithm natural exponential Exponential and Logarithmic Functions. |