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:

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

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

  3. 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: \(\tanh(-x) = - \tanh(x) \)
    Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely
    Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely
    Odd function: \(\tanh(-x) = - \tanh(x) \)
     Graph of hyperbolic tangent  function tanh(x)
    Fig.3 - Graph of Hyperbolic Tangent Function tanh(x)

  4. 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: \(\coth(-x) = - \coth(x) \)
    Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely
    Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely
    Vertical asymptote: \( x = 0 \)
    Odd function: \(\coth(-x) = - \coth(x) \)
     Graph of hyperbolic cotangent function coth(x)
    Fig.4 - Graph of Hyperbolic Cotangent Function coth(x)

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


    Domain: \( (-\infty , \infty) \)
    Range: \( (0 , 1] \)
    Even function: \(\text{sech}(-x) = \text{sech}(x) \)
    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)

  6. 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: \(\text{csch}(-x) = - \text{csch}(x) \)
    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

Differentiation of Hyperbolic Functions
Hyperbolic Functions Calculator

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