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 explored using an applet. The six hyperbolic functions are defined as follows:

  1. sinh(x) = (ex - e-x)/2

  2. cosh(x) = (ex + e-x)/2

  3. tanh(x) = sinh(x) / cosh(x) =  (ex - e-x) / (ex + e-x)

  4. coth(x) = cosh(x) / sinh(x) = (ex + e-x) / (ex - e-x)

  5. sech(x) = 1 / cosh(x) = 2 / (ex + e-x)

  6. csch(x) = 1 / sinh(x) = 2 / (ex - e-x)

If needed,
Free graph paper is available.

Interactive Tutorial

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  1. click on the button above "click here to start" and MAXIMIZE the window obtained.

  2. Click on the radio button of sinh (x). What is the domain of sinh (x)? Use the graph and other analytical tools, such the definitions above and the behavior of exponential functions in hint 1 below, to determine the range of sinh (x).
    Hint 1: ex approaches zero and e-x  increases without bound as x decreases without bound.  ex increases without bound and e-x approaches zero as x increases without bound.

  3. Click on the radio button of cosh (x). What is the domain of cosh (x)? Use analytical tools and the graph to determine the range of cosh (x).
    Hint 2 : consider x = 0 , x > 0 and x < 0

  4. Click on the radio button of tanh (x). What is the domain of tanh(x). Determine the range of tanh(x).

  5. Click on the radio button of coth(x). Is coth(x) defined at x = 0? What is the domain of coth(x). Is there a vertical asymptote at x = 0? Determine the range of coth(x).

  6. Click on the radio button of sech(x). What is the domain of sech(x). Determine the range of sech(x).

  7. Click on the radio button of csch(x). Is csch(x) defined at x = 0? Is there a vertical asymptote at x = 0? What is the domain of csch(x).Determine the range of csch(x).

    More references on hyperbolic functions

    Differentiation of Hyperbolic Functions

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Updated: February 2015

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