Problem 35 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ f(x)=\frac{8}{1+e^{-0.5 x}} $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{8}{1+e^{-0.5 x}}\) has one horizontal asymptote at \(y=8\).

Step 1: Analyze the Function

Identify the type and the features of the function. Here, \(f(x)=\frac{8}{1+e^{-0.5 x}}\) is a logistic function with a horizontal asymptote \(y=8\). The function has been transformed by stretching vertically by a factor of 8 and horizontally by a factor of -0.5.

Step 2: Graph the Function

Use a graphing utility to graph the function. Usually, this would involve inputting the function into the graphing utility and set the viewing window to display relevant features of the graph.

Step 3: Identify the Asymptotes

Observe the graph and identify the asymptotes. For logistic functions, the horizontal asymptote can usually be found at \(y = a\), where \(a\) is the numerator of the fraction in the logistic function.

Step 4: Confirm the Asymptotes

Look at the graph. As \(x\) approaches positive or negative infinity, the function's values should converge towards the asymptotes of \(y = 8\). This confirms our initial understanding of the function.

Key Concepts

Horizontal Asymptote

When dealing with a logistic function, understanding horizontal asymptotes is crucial. Horizontal asymptotes provide insight into the behavior of the graph as it stretches towards infinity. For the function given, \( f(x)=\frac{8}{1+e^{-0.5 x}} \), the horizontal asymptote exists at \( y=8 \). This is because, as \( x \) approaches either positive or negative infinity, the exponential term \( e^{-0.5 x} \) either grows extremely large or diminishes, pushing the function's value closer to a horizontal line.

• When \( x \to \infty \): The exponential, \( e^{-0.5 x} \), approaches zero, making \( f(x) \to 8 \).
• When \( x \to -\infty \): Conversely, \( e^{-0.5 x} \) becomes very large, causing \( f(x) \) to move towards zero.

This behavior is typical for logistic functions, where the horizontal asymptote directly provides a boundary that the function approaches but never quite reaches.

Graphing Utility

Utilizing a graphing utility is fundamental for visualizing complex functions like logistic functions. These tools are essential in exploring the function's nuances without manually plotting every point.To graph \( f(x)=\frac{8}{1+e^{-0.5 x}} \):

• Input the function into the graphing software.
• Set an appropriate viewing window on your graphing utility to capture the critical behaviors.

Adjusting the window can reveal nuances in function behavior, especially near asymptotes and transformation areas. For a logistic curve, ensure that your window provides a suitable range for the \( x \) and \( y \) axes to observe the function's approach towards its asymptote.

Function Transformation

Transformations provide a way to alter the graph of a basic function to better fit specific data or problem contexts. The logistic function \( f(x)=\frac{8}{1+e^{-0.5 x}} \) features several transformations that modify its original shape and position.

• Vertical Stretch: The whole function is multiplied by 8, stretching it vertically. As a result, every point on the graph is 8 times higher compared to the standard logistic graph \( \frac{1}{1+e^{-x}} \).
• Horizontal Stretch: The negative coefficient \(-0.5\) associated with \( x \) compresses the graph horizontally, effectively slowing the rate at which it approaches its asymptotes.

These transformations reshape the function, allowing it to model real-world phenomena more accurately. Understanding and visualizing these adjustments through graphing utilities can significantly aid in comprehending how logistic functions adapt under various transformations.