How do you find the equation of a rational function?

  1. 1. The problem statement, all variables and given/known data

    I have a picture of a rational function and I need help finding out the equation of the function.

    2. Relevant equations

    Vertical Asymptote/ (Bx+C)

    3. The attempt at a solution

    I know I have to find X and Y intercepts, which is what I did. But how do I find the vertical asymptote if I dont have the equation?

    I need to know the numerator of the function.
  2. jcsd
  3. Mark44

    Staff: Mentor

    Since you have the graph of the function, you should be able to figure out the vertical asymptotes from the vertical lines that separate the parts of the graph. One very simple rational function is f(x) = 1/x. The line x = 0 (the y-axis) is the vertical asymptote. Another simple rational function is g(x) = 1/(x^2), which also has the line x = 0 as a vertical asymptote. An important difference between these two functions is how the graph behaves on either side of the vertical asymptote. For the first function, the graph is sort of an upside-down U to the left of the VA, but has sort of the shape of a U to the right of the VA. For the second function, the graph has the same shape on both sides of the VA. The difference between these two functions is due to the multiplicity of the factor(s) in the denominator.

    Since you haven't provided an image of the graph of your function, there's not much more I can say about it.
  4. HallsofIvy

    HallsofIvy 40,218
    Staff Emeritus
    Science Advisor

    Your last sentence is "I need to know the numerator of the function." Vertical asymptotes really have no relevance to the numerator. If you know, from the graph or however you are given the function, that there is a vertical asymptote at x= x0, then you know that there (at least one) factor (x- x0) in the denominator. If x0 is a zero of the function, then you know there is (at least one) factor (x- x0) in the numerator. Of course, neither of those tells you about factors like x2[sup+ 1, which have no zeros, in the numerator or denominator.

    You can get some information by looking at the other asyptotes: if there is a horizontal asymptote, the degree of numerator and denominator is the same. If there is a slant, straight line, asymptote, then the degree of the numerator is 1 more than the degree of the denominator. If there is a parabolic asymptote, 2 more, etc.
  5. i dont know the functions, thats the problem.

    I know how to do everything once I know the function.

    My only problem is finding the function.
  6. But the point is you don't have the function. Without the picture or more information, we can't get more specific. You should know what an asymptote looks like from a picture without looking at the function. We can help you some more once you have found them.
  7. Horizontal Asy = -2.65

    Vertical Asy= -5
  8. HallsofIvy

    HallsofIvy 40,218
    Staff Emeritus
    Science Advisor

    Then what you were told in the first two responses is that you must have 1/(x+5)- 2.65. However there may be other terms that are not reflected in the asymptotes.
  9. Even though the question that I had to solve (creating a rational function knowing the intercepts, and some characteristics of the graph) was a little different from the question asked here, but reading your replies definitely helped me create a rational function. Thanks :)
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