The hyperbolic secant function is a mathematical function that characterizes the shape of hyperbolic curves.

Defined as the reciprocal of the hyperbolic cosine function, the hyperbolic secant function, denoted as $\text{sech}(x)$, can be expressed by the formula:

One of the notable properties of the hyperbolic secant function is that it is an even function, implying that it is symmetric with respect to the y-axis. Its range of values lies between $0$ and $1$. As $x$ approaches both positive and negative infinity, $\text{sech}(x)$ approaches $0$. The function reaches its maximum value of $1$ at $x = 0$.

The hyperbolic secant function finds applications in various fields, including statistics and physics, where it is used to describe the shape of certain curves, such as the probability density function of the standard normal distribution. Additionally, it plays a role in signal processing and control theory to characterize the response of specific systems.

To graph the hyperbolic secant function, one can utilize graphing calculators or software, or alternatively, plot points manually. For instance, consider the following points: $(-3, 0.099)$, $(-2, 0.265)$, $(-1, 0.648)$, $(0, 1)$, $(1, 0.648)$, $(2, 0.265)$, and $(3, 0.099)$. By creating a table of values and plotting these points on a graph, one will observe a symmetric hyperbolic curve that approaches the x-axis as $x$ tends towards both positive and negative infinity.