A hyperbola is a function in the form of xy = k or y =

This function is not defined when x=0, there will be a discontinuity at x=0, and y is .

When x is very large or very small, y becomes almost 0. Hence hyperbola is a discontinuous graph.

To sketch the graph of a hyperbola, we find the point of discontinuity and the asymptotes (lines which the curve approaches to meet but never touches).

The asymptotes for y = are the x-axis (when x is very large or very small), and the y-axis (when x = 0)

Domain and range of y = f(x) = are:

Domain: {all real x, except x=0}

Range: {all real y: y}

The function f(x) = is a hyperbola with asymptotes at x = and y axis (y)

Domain: {all real x, except x}

Range: {all real y: y}

The graph of the hyperbola f(x) = is translated by units towards the left along the x-axis from origin. The curve does not touch the line x = in f(x) = .

**Example 1**: Find the domain and range for the function y = f(x) =

This is an equation for hyperbola.

Domain: {all real x }. In other words x takes all real values except x = 2

Range: {all real y: y}

y-intercept is when x=0, so y = (0,)

also y cannot be zero (it never becomes 0 when x is very large or very small) x = , , y = . (0,)

The lines x = 2 and y = 0 (x-axis) are the lines where the graph/curve does not touch by approaches to touch it. x = 2 and the x-axis are vertical and horizontal asymptotes respectively.

**Example 2**: y = f(x) =

To find the domain, 3x + 6 y = f(x) = .

3x = -6, so x = -2

When x = -2, there is a discontinuity, or f(x) is undefined. So x = -2 is an asymptote.

To find the range y = f(x) . When x is very large or very small, f(x) approaches zero. So y = 0 is another asymptote, along with x-axis. So

Domain: {all real x, x}

Range: {all real y, y}

y-intercept – when x = 0; y = f(x) = = -2.

So the curve meets the y-axis at (0, -2).

**TO SUMMARISE**

To draw a hyperbola y =

- Find the value of x which gives the discontinuity. Therefore the line x = is an asymptote. Draw a dot line to represent it.
- Find what happens when x is very large or very small. x-axis is another asymptote.
- If k>0, the graph is in the first or third quadrant; and if k<0, the graph is the second and fourth quadrant
- Find y when x = 0 (y-intercept). This will help you do the graph accurately.

Remember the graph is almost touching x-axis and line x =