## 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 \(y = -\cfrac{1}{x}\).

\(a<0\) | \(a>0\) | |

\(q>0\) | ||

\(q=0\) | ||

\(q<0\) |