Math Example--Exponential Concepts--Graphs of Exponential Functions: Example 27

Topic

Exponential Functions

Description

This example illustrates the graph of the exponential function y = 4 * e^2x, where e is the mathematical constant approximately equal to 2.71828. The graph shows a rapidly increasing curve with a y-intercept at ( 0 , 4 ). Students are asked to graph the y-intercept and determine if the function is decreasing or increasing.

Exponential functions with base e are fundamental in various mathematical and scientific applications. By examining this example with both a coefficient multiplying the function and a coefficient in the exponent, students can develop a deeper understanding of how such modifications affect the behavior of natural exponential functions.

Providing students with a variety of worked-out examples is essential for building a comprehensive understanding of exponential functions. Each example highlights different aspects of these functions, allowing students to recognize patterns and make connections between different representations. This approach helps reinforce key concepts and improves students' ability to analyze and interpret exponential functions in various contexts.

Teacher Script: "In this twenty-seventh example, we have the natural exponential function multiplied by 4, and 2x in the exponent. How do these changes affect the graph's behavior compared to our previous examples? Can you explain why the function increases more rapidly than y = e^x or y = 4 * e^x? As we continue with more examples, try to predict how combinations of coefficients outside and inside the exponential expression will influence the graph's position and rate of increase."

For a complete collection of math examples related to Graphs of Exponential Functions click on this link: Math Examples: Graphs of Exponential Functions Collection.