Graphing a Function with Asymptotes
Date: 07/05/98 at 01:57:24 From: Nurul Huda Hasan Subject: Asymptotes of a function Dear Dr. Math, I'm having trouble solving problems regarding asymptotes of a function. What is an asymptote and how do you solve questions regarding this topic? I have tried to solve this one, but I need more guidance to solve this question. Thank you for your help. The question is: Find the asymptote for this function and draft the graph: 2x + 1 f(x) = ------ x - 3
Date: 07/07/98 at 12:07:57 From: Doctor Peterson Subject: Re: Asymptotes of a function Hi, Nurul. Asymptotes can be a little confusing because there are really three kinds, each of which needs a different technique. The general definition is that an asymptote is a line to which a curve approaches as you follow the curve to infinity. The difference is in how you go to infinity. First, a horizontal asymptote is a horizontal line of the form y = c. As x goes to infinity, if y approaches c as a limit, then y = c is an asymptote. An example would be: 2 y = 5 + ----- x - 3 which approaches: y = 5 as x goes to infinity, because (x - 3) goes to infinity and 2/(x-3) goes to zero. Second, a vertical asymptote is a vertical line of the form x = c. You approach this case inside-out: as x goes to c, if y goes to infinity then x = c is an asymptote. An example is again: 2 y = 5 + ----- x - 3 which approaches infinity as x goes to 3, so it has an asymptote: x = 3 Finally, any line y = m * x + b can be an asymptote. Here you can't technically just take the limit of the function as x goes to infinity, since the limit is infinite, but you can take the limit of the difference between your function and a line and see that this goes to zero. An example is: 2 y = 2*x + 5 + ----- x - 3 which approaches y = 2*x + 5 as x goes to infinity. Now how do you recognize an asymptote? In your case: 2*x + 1 y = ------- x - 3 I would first think informally about what happens when x goes to infinity, and where y might go to infinity. When x is very large, 2*x will be much bigger than 1, and x will be much bigger than 3, so you can ignore the 1 and 3, so (where =~ means approximately equal): 2*x y =~ --- = 2 x This suggests that you will have an asymptote y = 2. To prove it more carefully, you can use standard limit techniques: 2*x + 1 2 + 1/x 2 y = ------- = ------- --> - as x --> infinity x - 3 1 - 3/x 1 When will y be very large? Look in the denominator and ask when (x - 3) will be zero. This tells us to check what happens at x = 3: 2*x + 1 7 y = ------- = - = infinite x - 3 0 so there is another asymptote at x = 3. (If the numerator had also been zero, you would have had to look at the limit as x approaches 3.) Now you have one more thing to do: use this information to graph the function. I won't try to draw the graph for you, but what you need to do is to draw the asymptote lines y = 2 and x = 3, plot a couple easy points, such as where x is 1, 2, and 4 (you want some points near the vertical asymptote so you can see how it behaves there), and draw a curve that goes through those points and approaches the lines. I hope that helps you understand asymptotes better. If you have trouble with limits, write back and I'll see if I can help you the rest of the way. - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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