Finding Asymptotes

Date: 07/06/98 at 11:19:39
From: Gerald Robinson
Subject: Asymptotes

I have an equation of the form:

   y = (m/x) + c

where m and c are constants.

I think I understand how to get c but I am finding it difficult to get 
values for m graphically.

Also, how do I model a graph on this equation? I did this about three 
years ago at college, and now I have to use this sort of thing in 
training at work. I would be very grateful if you could give me some 
pointers.
 
Many thanks,
Gerald Robinson

Date: 07/06/98 at 11:38:25
From: Doctor Jaffee
Subject: Re: Asymptotes

Hi Gerald,

I think I can help you out. First of all, let's consider a specific 
example and then we'll generalize it.

Suppose that you know that m = 2 and c = 3. You need to examine the 
x- and y- asymptotes. There is no way that x could equal 0 in this 
equation because any number divided by 0 is undefined, so if you were 
to substitute 0 for x in the equation you would get y = 2/0 + 3, which 
has no value. Therefore, there must be a vertical asymptote along the 
y-axis because those are the points where x = 0.

Furthermore, there is no way that y could equal 3. If you substitute 
3 for y in the equation you will have 3 = 2/x + 3. Now subtract 3 from 
both sides and you get 0 = 2/x. But it is impossible to divide 2 by a 
number and end up with 0; therefore, there is no way that y can equal 
3. There must be a horizontal asymptote at y = 3.

If you start plotting points by picking values of x from -5 to 5, you 
will see that the graph will be a hyperbola with the one branch in the 
top right quadrant of the region formed by the two asymptotes. The 
other branch is in the lower left quadrant.

In general, then, the graph of y = m/x + c will be a hyperbola similar 
to the one I just discussed. The only exception would be if m were 
equal to 0. Then the equation would simplify to y = c and the graph 
would be a horizontal line. If m were negative the graph would be a 
hyperbola flipped upside down. The height of the horizontal asymptote 
depends on the value of c.

Finally, as the value of m changes you can see that the hyperbola will 
become more curvy or more L-shaped depending on the value of m.

I hope this explanation has helped. Write back again.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/06/98 at 11:49:41
From: Doctor Rob
Subject: Re: Asymptotes

Once you know c, find the point with y-coordinate c+1. The 
x-coordinate of that point will be m.

The way to graph any equation is to take a few representative values 
of x and use them to calculate the corresponding values of y. Each x 
and its corresponding y form the coordinates (x,y) of a point on the 
graph of the equation. Then plot the points (x,y) you get that way in 
the xy-plane. You will need more points near places where unusual 
things happen. In this case, you will need several points near x = 0, 
because there the value of y grows very large. Once you have plotted 
sufficient points, connect them up with a smooth curve.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/