Input boxes are used to change parameters included in the definition of the exponential function which in this tutorial has the form
The values of the coefficients a, b, c, d, and the base B may be changed continuously (small increments). This makes this interactive tutorial very helpful and leads to a deep understanding of the behavior of the graph of the exponential functions.
Definition of the Exponential Function
The basic exponential function is defined by
where B is the base such that B > 0 and B not equal to 1.
The domain of f is the set of all real numbers.
Example:

f(x) = 2^{x}

g(x) = 4^{x}

h(x) = 0.4^{x}

k(x) = 0.9^{x}
Exponential functions may be explored using an html5 applet (suitable for ipads and tablets) as shown below.
 Set a to 1, b to 1, c to 0, d to 0 and the base B to 2. This defines function f given in part a) in the example above. Zoom in and out if necessary. Read values from the graph and make sure that the graph you have corresponds to the function defined above. 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. This defines function g given in part b) in the example above. Again make sure that the graph corresponds to function g above. 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. This defines function h given in part c) in the example above. Again make sure that the graph corresponds to function g above. 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. This defines function h given in part c) in the example above. Again make sure that the graph corresponds to function k above. Does the graph of function f increase or decrease?
Answers to above questions.
Increase and Decrease of the Exponential Functions
Interactive Tutorial Using Java Applet (2)

Set a to 1, b to 1 , c to 0, d to 0 and change base B so that B > 1. Note that as long as B > 1, the exponential function B^{x} increases throughout its domain which is the set of all real numbers.

Set a to 1, b to 1 , c to 0, d to 0 and change base B so that 0 < B < 1. Note that as long as 0 < B < 1, the exponential function B^{x} decreases throughout its domain.
Range and Horizontal Asymptote of the Exponential Functions
Interactive Tutorial Using Java Applet (3)

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^{x} increases without bound (zoom in and out if necessary) and as x decreases B^{x} approaches zero but is never equal to zero. The graph follows the x axis. The range of B^{x} is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.

Set a to 1, b to 1, c and d to zero. Set base B to values smaller than 1 and note the following: as x decreases, B^{x} increases without bound (zoom in and out if necessary) and as x increases B^{x} approaches zero but is never equal to zero. The graph follows the x axis. The range of B^{x} is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.
Shifting, Scaling and Reflection of the Exponential Functions
We now investigate the effects of parameters a, b, c and d on the properties of the graph of function f defined by:
Interactive Tutorial Using Java Applet (4)
 Set B = e, b = 1, c = 0 and d = 0 and Explore the effects of parameter a (vertical scaling) on the graph of f.
 Set a = 1, c = 0, d = 0 and B = e and Explore the effects of parameter b (horizontal scaling) on the graph of f.
 Set a = 1, b = 1, d = 0 and B = e and Explore the effects of parameter c (horizontal shift) on the graph of f.
 set B,a,b,c to values of your choice, change d and explain how it affects the horizontal asymptote and the range of f.
 What parameter(s) affect the y intercept? Do you think the graph of this function will always have a y intercept? Explain analytically.
 What parameter(s) affect the x intercept? Do you think the graph of this function will always have an x intercept? Explain analytically.
Answers to above questions.
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 links to topics related to exponential functions.
