Identify Characteristics of Simple Rational Functions

Background

The hyperbola is one of the four conic sections (the others are the circle, the parabola and the ellipse) investigated by Appollonius of Perga over $2000$2000 years ago. Astronomers in the 17th and 18th century identified the curve as the path of a fast moving comet as it approaches the Sun. The shadow tip of a sun dial traces out an hyperbola, and the intersections of the ripples formed when two stones are thrown into a lake form hyperbolae also.  

Basic Form

The basic mathematical form of a rectangular hyperbola is given by $xy=a$xy=a, where $a$a is any non-zero real number (positive or negative) called the dilation constant.

Consider the following example given by $xy=12$xy=12. 

If two numbers $x$x and $y$y are chosen so that their product $12$12 remains constant, then it becomes clear that the choice of the first number $x$x will determine the choice of the second number $y$y.  If for example $x=1$x=1, then $y$y needs to be $12$12 so that $1\times12=12$1×12=12. If $x=2$x=2, then $y$y becomes $6$6 because $2\times6=12$2×6=12. Again, choosing $x=12$x=12 means $y=1$y=1. Thus the points $\left(1,12\right),\left(2,6\right)$(1,12),(2,6) and $\left(12,1\right)$(12,1) are three points on the hyperbola.

If $x$x is extremely small, say $x=\frac{1}{100}$x=1100​, then $y$y must be the extremely large number $1200$1200 so that $\frac{1}{100}\times1200=12$1100​×1200=12. The smaller $x$x gets the larger $y$y must be, and vice-versa. 

What about the case when $x$x is chosen as a negative number? In that case we know that $y$y must also be negative, so that the product of $x$x and $y$y remains $12$12. This means that points like $\left(-1,-12\right),\left(-2,-6\right),\left(-12,-1\right)$(−1,−12),(−2,−6),(−12,−1) and $\left(-\frac{1}{100},-1200\right)$(−1100​,−1200) are also on the curve. 

This inverse relationship between $x$x and $y$y (when $x$x is small, $y$y is large and when $x$x is large, $y$y is small) gives the hyperbola its characteristic shape shown here.

Asymptotes of a rectangular hyperbola

Note that the two arcs of the curve, in their extremities, move closer and closer to the axes, but never touch them. Lines that curves approach are known as asymptotes, so that the $x$x and $y$y axis are asymptotes of the hyperbola. We can say that as $x$x approaches $0$0, $y$y approaches infinity and when $x$x approaches infinity, $y$y approaches $0$0. 

Because the asymptotes are at right angles to each other, we refer to this hyperbola as a rectangular hyperbola. Many other forms of the hyperbola (with more complicated mathematical forms) have asymptotes that are not at right angles to each other. 

When the dilation constant is negative

When the constant $a$a is negative then $x$x and $y$y must always be of opposite sign. this means that the arcs of the hyperbola move across from quadrants $1$1 and $3$3 (where $x$x and $y$y are of the same sign) to quadrants $2$2 and $4$4 (where $x$x and $y$y is of different sign). Both asymptotes remain the same. Use the applet below to see the difference between the hyperbolae given by $xy=12$xy=12 and  $xy=-12$xy=−12. 

The effect of the dilation constant

The centre of the hyperbola is located at the origin (this is a similar idea to the centre of a circle). When $a$a becomes larger, the hyperbola $xy=a$xy=a appears to move away from its centre. When c is small the curve moves toward its centre. Use the applet to examine this effect. Try large and small values of $a$a, and check that the same effect occurs when $a$a is negative. 

In particular, the curve $xy=1$xy=1  (often written as $y=\frac{1}{x}$y=1x​) with $a=1$a=1, has every point with either $x$x within the interval $-1\le x\le1$−1≤x≤1 or $y$y within the interval $-1\le y\le1$−1≤y≤1. Use the applet to see how close the two arcs are to the axes. 

Note that the location of the asymptotes are not affected by the size of $a$a. 

The hyperbola as a function 

As a function, $xy=a$xy=a can be written as $y=\frac{a}{x}$y=ax​, so that $x$x is identified as the independent variable. 

In the new form, the hyperbola $y=\frac{a}{x}$y=ax​ can be written functionally as $f\left(x\right)=\frac{a}{x}$f(x)=ax​.

So for example $f\left(x\right)=\frac{12}{x}$f(x)=12x​ shows $f\left(1\right)=\frac{12}{1}=12,f\left(2\right)=\frac{12}{2}=6$f(1)=121​=12,f(2)=122​=6 and $f\left(-\frac{1}{100}\right)=\frac{12}{\left(-\frac{1}{100}\right)}=-1200$f(−1100​)=12(−1100​)​=−1200.

 

Applet

This applet allows you to explore some simple forms of the basic hyperbola.  Notice what happens to the end behaviour and asymptotes as you change the function.   

 

 

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3