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Finding the equation of a graph with asymptotes (help please) 
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#1
Feb2009, 11:38 PM

P: 2

1. The problem statement, all variables and given/known data
This is the question: Shown in the figure below is the graph of a rational function with vertical asymptotes x=2, x=6, and horizontal asymptote y= 2 . (All xintercepts of the graph of f are also shown, and a point on the graph is indicated.) The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x) , and then write the equation. I can't get the graph on but point are at (4,0); (0,2); (3,0). 2. Relevant equations The answers choices are: A) f(x)= a / xb B) f(x)= a(xb) / xc C) f(x)= a / (xb)(xc) D) f(x)= a(xb) / (xc)(xd) E) f(x)= a(xb)(xc) / (xd)(xe) 3. The attempt at a solution What I first did was try and get rid of equations that couldn't possible work. So I knew I had vertical asymptotes at x=2,6. I had two asymptotes so I knew that options A and B couldn't work. I then got rid of option C because dividing a by x to get my horizontal asymptote would not give me y= 2 it would give me y=0. I then got rid of option E by factoring the top part through. Since having ax^2 as my leading coificent would give me a diagonal asymptote I got rid of it. So I got D as my answer but then I have to find the equation. So I got (x2)(x6) for the bottom because those would give me vertical asymptotes at x=2,6 But I don't know how to find the top. I got 2(x12) divided by (x2)(x6) because sticking 0 in for x would give me 2. So I had the point (0,2) but I can't get any of the others to match up. I'm not sure how to figure it out. 


#2
Feb2109, 04:55 AM

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You are right, that the other points don't match up. To put it in a mathematical way: the set of equations for a, b, c... you get from plugging in the points is overdetermined and does not have a solution. Therefore D is wrong.
I think your argument for getting rid of F is not entirely correct. You are saying, that your leading behaviour is ~ x^2. However, you also have something ~ x^2 in the denominator. Did you learn to take limits yet? Can you properly calculate [tex]\lim_{x \to \infty} \frac{a (x  b)(x  c) }{ (x  d)(x  c) }[/tex] Also note that you will need all the points to determine the constants. To take your (wrong) example of option D: if you plug in x = 0 then you get f(0) =  a b / 12 = 3. Although you get a relation a = 36 / b, this alone will not allow you to determine a and b, you will need at least one other equation. 


#3
Feb2109, 05:45 AM

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Does option F really have "(x c)" in both numerator and denominator? And what happened to option E?



#4
Feb2109, 06:35 AM

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Finding the equation of a graph with asymptotes (help please)



#5
Feb2109, 01:22 PM

P: 2

No I really don't know how to do limits. We've touched on it a little bit last year. But I am in an online precalculus course and the program doesn't really explain things in terms I can understand. The program has never shown me how to do this so I have no idea how to do it. I'm completly confused and lost. I'm slow at learning math so most of the time I need it in the simplest terms possible.



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