Mathematics » Functions II » Hyperbolic Functions

Revision of Hyperbolic Functions

Functions of the form \(y=\cfrac{a}{x}+q\)

Functions of the general form \(y = \cfrac{a}{x} + q\) are called hyperbolic functions, where \(a\) and \(q\) are constants.

The effects of \(a\) and \(q\) on \(f(x) = \cfrac{a}{x} + q\):

  • The effect of \(q\) on vertical shift

    • For \(q>0\), \(f(x)\) is shifted vertically upwards by \(q\) units.

    • For \(q<0\), \(f(x)\) is shifted vertically downwards by \(q\) units.

    • The horizontal asymptote is the line \(y = q\).

    • The vertical asymptote is the \(y\)-axis, the line \(x = 0\).

  • The effect of \(a\) on shape and quadrants

    • For \(a>0\), \(f(x)\) lies in the first and third quadrants.

    • For \(a > 1\), \(f(x)\) will be further away from both axes than \(y = \cfrac{1}{x}\).

    • For \(0<a<1\), as \(a\) tends to \(\text{0}\), \(f(x)\) moves closer to the axes than \(y = \cfrac{1}{x}\).

    • For \(a<0\), \(f(x)\) lies in the second and fourth quadrants.

    • For \(a < -1\), \(f(x)\) will be further away from both axes than \(y = – \cfrac{1}{x}\).

    • For \(-1<a<0\), as \(a\) tends to \(\text{0}\), \(f(x)\) moves closer to the axes than \(y = -\cfrac{1}{x}\).

 

\(a<0\)

\(a>0\)

\(q>0\)

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\(q=0\)

45839a0b1faab23d0b3c5bf15e28c149.png61c1d3c6221dbfb749b6dd501313928b.png

\(q<0\)

686c6c6d1a05261f600bdb46b2701b4e.png58497288c3d81a47c1125eb97ad2e6ab.png

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