Question

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

Short answer

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

Step-by-step solution

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.

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.

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.

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.