Exponential Functions and Compound Interest Study Guide
This comprehensive study guide explores the fundamental concepts of exponential functions and compound interest, providing essential information for students learning about growth and decay patterns in mathematics and finance.
Definition: Exponential functions are mathematical expressions based on multiplication patterns rather than addition.
The guide begins by emphasizing the crucial difference between exponential and linear functions, highlighting that exponential growth and decay are characterized by multiplicative changes.
Highlight: Exponential growth occurs when the multiplier (b) in the function y = ab^n is greater than 1 (b > 1).
For exponential growth, the guide provides an example: y = 2(2)^x, illustrating how the function increases over time. This concept is directly related to appreciation in financial contexts.
Example: In financial terms, if an asset appreciates by 6%, the multiplier used in the exponential function would be b = 1.06.
The study material also covers exponential decay, which occurs when the multiplier is between 0 and 1 (0 < b < 1). An example given is y = 2(0.5)^x, demonstrating how the function decreases over time. This concept is linked to depreciation in economics.
Example: For a 6% depreciation rate, the multiplier in the exponential function would be b = 0.94.
The guide provides the general formula for exponential functions: y = ab^n, where 'a' represents the starting value, 'b' is the base (or multiplier), and 'n' is the exponent (often representing time).
Vocabulary: In the context of finance, 'appreciation' refers to an increase in value over time, while 'depreciation' indicates a decrease in value.
The material distinguishes between simple interest and compound interest, noting that compound interest follows an exponential pattern. It also touches on linear functions (y = mx + b) to contrast with exponential growth.
Highlight: When given two points on a graph, students are advised to use a system of equations to solve for the variables in the exponential function.
This study guide serves as a valuable resource for understanding exponential growth and decay examples with answers, providing a solid foundation for more advanced topics in mathematics and financial modeling.
