Home > Introduction to Pre-Calculus > Introduction to Graphing Functions > Hyperbola


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

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

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 = k/x 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) = 1/x are:

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

Range: {all real y: y<>0}

The function f(x) = a/{bx~+~c} is a hyperbola with asymptotes at x = -c/b and y axis (y<>0)

Domain: {all real x, except x<>~-c/b}

Range: {all real y: y<>0}

The graph of the hyperbola f(x) = 1/x is translated by -c/b units towards the left along the x-axis from origin. The curve does not touch the line x = -c/b in f(x) = a/{bx~+~c}.

Example 1: Find the domain and range for the function y = f(x) = 5/{x~-~2}

This is an equation for hyperbola.

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

Range: {all real y: y<>0}

y-intercept is when x=0, so y = <>~-5/2      (0,<>~-5/2)

also y cannot be zero (it never becomes 0 when x is very large or very small) x = <>infty{y}right{0}, y = <>~-5/2.                   (0,<>~-5/2)

hyperbola example1

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) = -12/{3x~+~6}

To find the domain, 3x + 6 y = f(x) = <>~0.

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) <>~0. 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<>~-2

Range: {all real y, y<>~0}

y-intercept – when x = 0; y = f(x) = -12/{0~+~6} = -2.

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

hyperbola example2


To draw a hyperbola y = k/{ax~+~b}

  1. Find the value of x which gives the discontinuity. Therefore the line x = -b/a is an asymptote. Draw a dot line to represent it.
  2. Find what happens when x is very large or very small. x-axis is another asymptote.
  3. If k>0, the graph is in the first or third quadrant; and if k<0, the graph is the second and fourth quadrant
  4. 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 = -b/a