Graphs of Hyperbolic Functions

The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are presented. The six hyperbolic functions are defined as follows:

    Hyperbolic Sine Function : \( \sinh(x) = \dfrac{e^x - e^{-x}}{2} \)

    The graph of \( y = \sinh(x) \) is shown below along with the graphs of \( y = \dfrac{e^x}{2} \) and \( y = - \dfrac{e^{-x}}{2} \) for comparison

    Domain: \[ (-\infty , \infty) \] Range: \[ (-\infty , \infty) \] Odd function since: \(\sinh(-x) = - \sinh(x) \)

    Graph of hyperbolic sine function sinh(x)
    Fig.1 - Graph of Hyperbolic Sine Function sinh(x)

    Hyperbolic Cosine Function : \( \cosh(x) = \dfrac{e^x + e^{-x}}{2} \)

    The graph of \( y = \cosh(x) \) is shown below along with the graphs of \( y = \dfrac{e^x}{2} \) and \( y = \dfrac{e^{-x}}{2} \) for comparison

    Domain: \[ (-\infty , \infty) \] Range: \[ [1 , \infty) \] Even function since: \(\sinh(-x) = \sinh(x) \)

    Graph of hyperbolic cosine function cosh(x)
    Fig.2 - Graph of Hyperbolic Cosine Function cosh(x)

    Hyperbolic Tangent Function : \( \tanh(x) = \dfrac{\sinh(x)}{\cosh(x)} = \dfrac{e^x - e^{-x}}{e^x + e^{-x}} \)

    Domain: \[ (-\infty , \infty) \] Range: \[ (-1 , 1) \] Odd function since: \( \tanh(-x) = - \tanh(x) \} \)

    Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely

    Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely

    Odd function since: \(\tanh(-x) = - \tanh(x) \)

    Graph of hyperbolic tangent function tanh(x)
    Fig.3 - Graph of Hyperbolic Tangent Function tanh(x)

    Hyperbolic Cotangent Function : \( \coth(x) = \dfrac{\cosh(x)}{\sinh(x)} = \dfrac{e^x + e^{-x}}{e^x - e^{-x}} \)

    Domain: \[ (-\infty , 0) \cup (0, \infty) \] Range: \[ (-\infty , -1) \cup (1 , \infty) \] Odd function since: \( \coth(-x) = - \coth(x) \)

    Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely

    Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely

    Vertical asymptote: \[ x = 0 \] Odd function since: \(\coth(-x) = - \coth(x) \)

    Graph of hyperbolic cotangent function coth(x)
    Fig.4 - Graph of Hyperbolic Cotangent Function coth(x)

    Hyperbolic Secant Function : \( \text{sech}(x) = \dfrac{1}{\cosh(x)} = \dfrac{2}{e^x + e^{-x}} \)

    Domain: \[ (-\infty , \infty) \] Range: \[ (0 , 1] \] Even function since: \( \text{sech}(-x) = \text{sech}(x) \) Horizontal asymptote: \( y = 0 \) as \( x \) increases or decreases indefinitely

    Graph of hyperbolic secant function sech(x)
    Fig.5 - Graph of Hyperbolic Secant Function sech(x)

    Hyperbolic Cosecant Function : \( \text{csch}(x) = \dfrac{1}{\sinh(x)} = \dfrac{2}{e^x - e^{-x}} \)

    Domain: \[ (-\infty , 0) \cup (0 , \infty) \] Range: \[ (-\infty , 0) \cup (0 , \infty) \] Odd function since : \[ \text{csch}(-x) = - \text{csch}(x) \] Horizontal asymptote: \( y = 0 \) as \( x \) increases or decreases indefinitely

    Vertical asymptote: \( x = 0 \)

    Graph of hyperbolic cosecant function csch(x)
    Fig.6 - Graph of Hyperbolic Cosecant Function csch(x)
More references on hyperbolic functions