# Hyperbolic Function

A hyperbola is a function with its equation in the form $xy=a$ or $y=\cfrac{a}{x}$.

For example, $y=\cfrac{1}{x}$ is a discontinuous curve since the function is undefined at x=0.

Drawing up the table of values gives:

Reflection:

What happens to the graph as x becomes closer to 0?

What happens as x becomes very large in both positive and negative directions?

The value of y is never 0. Why?

To sketch the graph of a hyperbola, we can use the domain and range to help find the asymptotes (lines towards which the curve approaches but never touches.

Use this Geogebra file to explore the hyperbolic function:

### Examples:

1. a) Find the domain and range of $f\left( x \right) =\cfrac { 3 }{ x-3 }$.

b) Hence sketch the graph of the function.

This is the equation of a hyperbola. To find the domain, we notice that

$x-3\neq 3$

So, $x\neq 3$

Also, y cannot be zero.

Domain: {all real x: x $\neq$3}

Range: {all real y: y $\neq$0}

The lines x=3 and y=0 (the x-axis) are called the asymptotes.

To make the graph more accurate we can find another point or two. The easiest one to find is the y-intercept. For y-intercept, x=0.

$y=\cfrac { 3 }{ 0-3 } \\ \\ y\quad =\quad -1$

2. Sketch $y=\cfrac { 1 }{ 2x+4 }$.

This is the equation of a hyperbola. The negative sign turns the hyperbola around so that it will be in the opposite quadrants. If you are not sure where it will be, you can find two or three points on the curve. To find the domain, we notice that $2x+4\neq 0$.

$2x\neq 4\\ x\neq -2$

For the range, y can never be zero.

Domain: {all real x:$x:x\neq -2$}

Range: {all real y:$y:y\neq 0$

So there are asymptotes at x = -2, and = 0.

To make the graph more accurate, we can find the y-intercept. For y-intercept, x=0:

$y=-\cfrac { 1 }{ 2(0)+4 } \\ \\ y=-\cfrac { 1 }{ 4 }$

Therefore, we can summarise the hyperbolic functions with the following rule: