Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 42

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = -7 * e^-x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating how the function's output changes as x increases. The accompanying graph provides a visual representation of this exponential decay function with a negative coefficient, highlighting its distinctive shape and behavior.

Exponential functions with negative exponents and coefficients are fundamental in mathematics and have numerous real-world applications, such as modeling radioactive decay, cooling processes, and certain economic phenomena. This collection of examples aids in teaching the topic by offering students diverse representations of exponential functions, enabling them to observe how different parameters influence the function's behavior, particularly when dealing with negative coefficients and exponents.

The importance of multiple worked-out examples in understanding complex exponential functions cannot be overstated. Each example in this set showcases a unique form of the exponential function, helping students identify patterns and comprehend how alterations in the coefficient and exponent affect the function's graph and values. This variety of examples strengthens the concept and assists students in developing a more intuitive grasp of exponential behavior, especially in cases where the function exhibits decay and is reflected across the x-axis.

Teacher Script: "Let's examine the function y = -7 * e^-x. Notice how this function decreases rapidly as x increases, starting from a negative y-intercept at (0, -7). The negative coefficient not only flips the graph vertically but also stretches it, while the negative exponent causes the function to decay. Compare this to our previous example with y = -1 * e^-x. How does the larger negative coefficient affect the graph's shape and y-intercept? Can you think of any real-world scenarios where we might encounter such a rapidly decaying negative exponential function?"

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