Finding AsymptotesDate: 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/ |
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