Problem 125 - Elementary and Intermediate Algebra

Question

Use a graphing calculator to graph the function \(f(x)=\frac{1}{1+e^{-2 x}} .\) Describe its graph in words.

Step-by-Step Solution

Answer

The graph is an S-shaped curve with horizontal asymptotes at \(y=0\) and \(y=1\), crossing the y-axis at \(0, 0.5\).

Step 1: Understanding the Function

The function given is a logistic function, commonly used to model S-shaped growth called a sigmoid curve. The function is defined as follows: \[ f(x) = \frac{1}{1 + e^{-2x}} \] It demonstrates how the function approaches horizontal asymptotes as \(x\) moves towards positive or negative infinity.

Step 2: Identify Key Features

To graph the function, first note the key features:1. Horizontal asymptotes at \(y = 0\) and \(y = 1\).2. The function is symmetric about the origin.3. It intersects the y-axis at \((0, 0.5)\) because when \(x = 0\), \[ f(0) = \frac{1}{1 + e^0} = \frac{1}{2}. \]

Step 3: Plot Critical Points

Using the graphing calculator, plot critical points for the function by selecting several \(x\) values and calculating \(f(x)\):- \(f(-2) \approx 0.12\)- \(f(-1) \approx 0.27\)- \(f(0) = 0.5\)- \(f(1) \approx 0.73\)- \(f(2) \approx 0.88\)These points help determine the shape of the graph.

Step 4: Sketch the Graph

On the graphing calculator, input the function and plot the calculated points. Notice the curve starts near zero, increases steeply near \(x=0\), and then levels off near 1 as \(x\) increases. This characteristic S-shape is typical of logistic functions.

Step 5: Describe the Graph

The graph of the function is an S-shaped curve. It starts at the horizontal asymptote \(y = 0\), rises steeply through the midpoint \((0, 0.5)\), and then flattens towards the horizontal asymptote \(y = 1\) as \(x\) increases. The curve is symmetric about the point \((0, 0.5)\).

Key Concepts

Logistic Function

A logistic function is quite intriguing and has some special characteristics. It is defined by the equation \[ f(x) = \frac{1}{1 + e^{-2x}} \]. In this equation, \(e\) is the base of the natural logarithm, approximately equal to 2.71828. Logistic functions are frequently used to model populations, biological systems, or any phenomena that start slowly, increase rapidly, and then level off. They feature a characteristic S-shaped curve, often referred to as a sigmoid curve. This function has a range that is typically between 0 and 1. Logistic functions are also versatile because they can continuously transition from one level to another in a smooth and predictable fashion, unlike a step function that changes abruptly.

• Growth starts gradually, exhibiting slow changes at first.
• The middle section shows exponential growth, where changes happen rapidly.
• Growth slows down again as it approaches its upper limit.

Horizontal Asymptote

Horizontal asymptotes play a key role in understanding the behavior of logistic functions. An asymptote is a line that a graph approaches but never actually touches. For the logistic function \[ f(x) = \frac{1}{1 + e^{-2x}} \], it has two horizontal asymptotes: one at \( y = 0 \) and another at \( y = 1 \). When \( x \) moves towards negative infinity, the function approaches the lower asymptote at \( y = 0 \). This occurs because the exponential function \( e^{-2x} \) becomes very large and the denominator increases significantly, leading to a smaller value for \( f(x) \).
On the other hand, as \( x \) moves towards positive infinity, \( e^{-2x} \) becomes very small. In this scenario, the denominator approaches 1, and thus \( f(x) \) tends towards the upper asymptote at \( y = 1 \).

• Asymptote at \( y = 0 \) when \( x \to -\infty \)
• Asymptote at \( y = 1 \) when \( x \to \infty \)

Understanding these asymptotes helps us visualize where the logistic function stabilizes.

Sigmoid Curve

The sigmoid curve is a fascinating graphical representation of logistic functions. Imagine it like an elongated "S" stretching across both directions. This kind of curve represents a gradual transition from one horizontal asymptote to another, reflecting various stages of growth or decline. In the context of the logistic function \( f(x) = \frac{1}{1 + e^{-2x}} \), the sigmoid curve expresses how the value starts low, sharply increases in the middle, and levels off high. The curve is symmetric around the central point \( (0, 0.5) \), displaying the same behavior on both sides.
Key characteristics of a sigmoid curve include:

• Starts and levels off gradually, creating soft edges at both asymptotes.
• Features a rapid growth span or transition in the middle.
• Is symmetric, making it predictable and uniform.

The beauty of the sigmoid curve is how elegantly it captures processes with initial hesitance that gain momentum before stabilizing once more.