Describe the hyperbolic secant function.

The hyperbolic secant function is a mathematical function that describes the shape of a hyperbolic curve.

The hyperbolic secant function, also known as sech(x), is defined as the reciprocal of the hyperbolic cosine function, cosh(x). It is given by the formula:

sech(x) = 1/cosh(x)

The hyperbolic secant function is an even function, which means that it is symmetric about the y-axis. It has a range of values between 0 and 1, and approaches 0 as x approaches infinity or negative infinity. The function has a maximum value of 1 at x = 0.

The hyperbolic secant function is commonly used in statistics and physics to describe the shape of certain curves, such as the probability density function of the standard normal distribution. It is also used in signal processing and control theory to describe the response of certain systems.

To graph the hyperbolic secant function, we can use a graphing calculator or software, or we can plot points by hand. For example, to plot the points (-3, 0.099), (-2, 0.265), (-1, 0.648), (0, 1), (1, 0.648), (2, 0.265), and (3, 0.099), we can use a table of values and plot them on a graph. The resulting graph will be a symmetric hyperbolic curve that approaches the x-axis as x approaches infinity or negative infinity.