Math Example--Exponential Concepts--Graphs of Exponential Functions: Example 24

Topic

Exponential Functions

Description

This example illustrates the graph of the exponential function y = e^0.5x, where e is the mathematical constant approximately equal to 2.71828. The graph shows an increasing curve with a y-intercept at ( 0 , 1 ). 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, including calculus, physics, and engineering. By examining this example with a fractional 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-fourth example, we have the natural exponential function with 0.5x in the exponent. How does this change the graph's behavior compared to our previous examples with whole number coefficients? Can you explain why the function increases more slowly than y = e^x? As we continue with more examples, try to predict how fractional coefficients in the exponent will influence the graph's 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.