21.09 The hyperbola

The hyperbola

The graph of an inverse relationship in the $xy$xy-plane is called a hyperbola. Let's see what inverse variation looks like in a table of values.

This table shows the relationship $y=\frac{1}{x}$y=1x​:

Notice that:

• If $x$x is positive, $y$y is positive
• If $x$x is negative, $y$y is negative
• As $x$x gets further from zero (in either direction), $y$y gets closer to zero

Here are some hyperbolas with equations of the form $y=\frac{k}{x}$y=kx​ (or $xy=k$xy=k).

Notice the following features:

• Each hyperbola has two parts and they lie in opposite quadrants.
• They all approach the line $y=0$y=0 (the $x$x-axis), and they also approach the line $x=0$x=0 (the $y$y-axis), but they never cross these lines. This is because the equation $y=\frac{k}{x}$y=kx​ is not defined for $x=0$x=0 or $y=0$y=0. We call the lines $x=0$x=0 and $y=0$y=0 asymptotes.
• They are symmetrical about the lines $y=x$y=x and $y=-x$y=−x, and they have rotational symmetry about the origin.
• We can see that every $x$x and $y$y coordinate on a hyperbola multiply to give the value of $k$k in their specific equation, as is suggested by the general form $xy=k$xy=k. (This gives us an easy method of finding an equation of a hyperbola from a sketch.)

Note: In this course you can use technology to graph hyperbolas, find stationary points, intercepts, and points of intersection.

Practice Questions

Question 1

Question 2