Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 29

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = -0.5 * 10^-0.3x. The image displays both a graph and a corresponding table, illustrating the behavior of this negative exponential function with a fractional coefficient and a negative fractional exponent. The table includes key points such as (-2, -1.99), (-1, -0.998), (0, -0.5), (1, -0.251), and (2, -0.126), demonstrating how the y-values approach zero from the negative side as x increases.

Exponential functions are a vital concept in mathematics, with applications in various fields such as physics, chemistry, and economics. This collection of examples helps teach the topic by providing visual and numerical representations of different types of exponential functions, including those with negative coefficients, fractional coefficients, and negative fractional exponents. Students can observe how these factors affect the graph's shape, the y-intercept, the rate of change, and the overall behavior of the function, developing a more comprehensive understanding of exponential relationships.

Offering multiple worked-out examples is essential for students to fully grasp the concept of exponential functions. By examining diverse cases, students can identify patterns, compare and contrast different functions, and develop a deeper insight into how exponential functions behave in various scenarios. This approach helps reinforce key concepts and improves students' ability to analyze and interpret exponential relationships in different contexts, such as radioactive decay, sound intensity, or compound interest with continuous compounding.

Teacher Script: "Let's examine this interesting exponential function, y = -0.5 * 10^-0.3x. How does it differ from our previous examples? Notice the negative fractional coefficient -0.5 and the negative fractional exponent -0.3x. How do these affect the graph's shape and the rate at which the y-values approach zero? Can you describe what happens to the y-values as x increases? This function demonstrates a slow exponential decay in the negative direction. How might we use this type of function to model real-world situations, such as the gradual dissipation of a pollutant in the environment?"

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