Question

In Exercises, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ g(x)=\frac{10}{1+e^{-x}} $$

Short answer

To graph the logistic function \(g(x) = \frac{10}{1+e^{-x}}\), use a suitable graphing utility and set the viewing window to include at least the x-values between -10 and 10 and y-values between 0 and 10. The curve should have an 'S' shape, approaching y = 0 and y = 10 without touching them, and have an inflection point around x = 0, y = 5.

Step-by-step solution

Understanding the function

The function \(g(x) = \frac{10}{1+e^{-x}}\) is a logistic function. The graph of logistic functions basically have an 'S' shape. The y-values of the function increase slowly at first, then more rapidly in the middle, and then slow down again as x increases.

Using a graphing utility

To graph this function, use a graphing calculator or a graphing utility online such as Desmos. Start by entering the function \(g(x) = \frac{10}{1+e^{-x}}\) into the utility.

Choosing an appropriate viewing window

The viewing window should include a range of x-values that allows the entire S-shape of the graph to be seen. A range of -10 to 10 on the x-axis and a range of 0 to 10 on the y-axis would be a good starting point. Adjust the window as necessary until the entire 'S' shape is visible and the key features of the graph can be clearly distinguished.

Analyzing the graph

The curve of the graph should approach the lines y = 0 and y = 10, but never touch or cross these lines due to the nature of the logistic function. The point at which the curve changes from increasing slowly to increasing more rapidly is called the inflection point and this should be approximately at x = 0, y = 5.

Key concepts

Logistic Function

A logistic function, such as the one given by the equation \(g(x) = \frac{10}{1+e^{-x}}\), describes growth that increases quickly at first, before slowing down and approaching an upper limit. The logistic function is often used to model population growth, chemical reactions, and certain types of data in social sciences. In such models, the function represents the idea that growth cannot continue indefinitely at an exponential rate due to limiting factors such as resources or space. The upper limit, in the case of our function, is the horizontal asymptote at \(y = 10\).

The interesting feature of logistic functions is the 'S' shape of their graph. This shape reflects the rate of growth: initially slow, then rapid, and finally slow again as it approaches the upper limit. The point in the middle of the 'S' curve where the growth rate switches from increasing to decreasing is called the inflection point. Understanding these characteristics is essential for interpreting and applying logistic models.

Using Graphing Utilities

Graphing utilities, whether software or online tools like Desmos, are invaluable for visualizing mathematical functions. To graph a logistic function using one of these tools, simply input the function's equation. In the case of \(g(x) = \frac{10}{1+e^{-x}}\), you would type it exactly as shown into the utility's interface.

The utility will plot the points and draw the curve, rendering a visual representation of the equation. This visual aid helps in understanding the behavior of the function, identifying the inflection point, and observing the horizontal asymptotes. The graphing utility brings abstract equations to life, making the concepts more concrete and easier to grasp.

Choosing Viewing Window for Graphs

Selecting an appropriate viewing window is critical when graphing functions, especially for those new to using graphing utilities. Your first task is to predict the relevant range of the x-values and the limits of the y-values. For the logistic function \(g(x) = \frac{10}{1+e^{-x}}\), a good starting view window might be \(-10 \le x \le 10\) and \(0 \le y \le 10\).

The goal is to capture the complete 'S' shape of the curve, ensuring you can clearly see the approach to both the lower and upper bounds of the function. The choice of window can affect the understanding of the graph's behavior, so adjust it as necessary to get an accurate picture of the function's properties. If you're uncertain, start with a broader view and refine it to focus on interesting regions of the graph, such as where the curve steepens or flattens out.

Properties of Logistic Functions

The logistic function has several key properties. Firstly, the graph is always contained within a pair of horizontal asymptotes, which means there are limits the function will never exceed. In the case of our \(g(x)\), these limits are \(y = 0\) and \(y = 10\).

Secondly, there is always an inflection point where the rate of growth changes. For the given logistic function, this occurs around the middle of the 'S' curve. Typically, the x-coordinate of the inflection point can be found where \(g'(x) = g''(x) = 0\), providing a crucial insight into the function’s growth dynamics.

Lastly, logistic functions are symmetrical about the inflection point. Understanding these properties not only aids in graphing the function but also in interpreting the significance of the function in real-world scenarios where logistic growth occurs.