Exponential Function

An exponential function is a mathematical function of the form:

$$ f(x) = a \cdot b^x $$

Where:

• $$ a $$ is the initial value or the value of the function when $$ x = 0 $$
• $$ b $$ is the base of the exponential function, often referred to as the growth factor when $$ b > 1 $$ or decay factor when $$ 0 < b < 1 $$
• $$ x $$ is the exponent

Properties of Exponential Functions

1. Asymptotic Behavior: Exponential functions have an asymptote at $$ y = 0 $$ for growth functions ($$ b > 1 $$) and at $$ y = 0 $$ for decay functions ($$ 0 < b < 1 $$).

2. Growth and Decay: When $$ b > 1 $$, the function exhibits exponential growth. When $$ 0 < b < 1 $$, the function exhibits exponential decay.

3. Domain and Range: The domain of an exponential function is $$ (-\infty, \infty) $$ and the range depends on whether the function represents growth or decay.

Graphing Exponential Functions

To graph an exponential function, you can use the following steps:

1. Identify the initial value $$ a $$ and the base $$ b $$
2. Plot the initial point $$ (0, a) $$
3. Use the base $$ b $$ to determine additional points by choosing values of $$ x $$ and calculating $$ y = a \cdot b^x $$
4. Connect the points to form a smooth curve

Key Concepts to Remember

• Exponential functions can model growth or decay phenomena
• The base of the exponential function determines whether it represents growth or decay
• Exponential functions have specific domain and range properties

Now, you can use this study guide to practice and master the concept of exponential functions!

[Exponential Function] Related Worksheets and Study Guides:

• FunctionsMathematics • Eighth Grade