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

  • 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

	\(a<0\)	\(a>0\)
\(q>0\)
\(q=0\)
\(q<0\)