Math Example--Exponential Concepts--Graphs of Exponential Functions: Example 22

Topic

Exponential Functions

Description

This example illustrates the graph of the exponential function y = 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 , 1 ). Students are asked to graph the y-intercept and determine if the function is decreasing or increasing.

Exponential functions with base e, known as natural exponential functions, play a crucial role in various mathematical and scientific applications, including calculus, physics, and engineering. By examining this example with 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 multiple worked-out examples is essential for mastering the concept of exponential functions. Each example introduces a unique combination of coefficients and exponents, allowing students to observe how these elements interact to produce different graph characteristics. This approach helps students build intuition and analytical skills necessary for working with exponential functions in diverse contexts.

Teacher Script: "Let's examine our twenty-second example. We have the natural exponential function, but now with 2x in the exponent. How does this change the graph's behavior compared to y = e^x? Can you explain why the function increases more rapidly? As we continue exploring more functions with base e, try to predict how different coefficients in the exponent will affect the steepness of the curve."

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