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Math Help - How do I find the asymptotes of this rational function?

  1. #1
    s3a
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    How do I find the asymptotes of this rational function?

    How do I find the asymptotes of:

    y = -(x-8)/(x+4)

    I am assuming that it is (h,k) in f(x) = a/b(x-h) + k but there is no k meaning it should equal to zero however that is not one of the multiple choice answers for the horizontal asymptote. The vertical asymptote is the value of h, right? Meaning it should be - 4, right?

    Assuming that I am right for the vertical asymptote, someone please help me find the horizontal one.

    Any help would be GREATLY appreciated!
    Thanks!
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  2. #2
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    y = \frac{-(x-8)}{x+4} = \frac{-x+8}{x+4}

    To find the vertical asymptote, take the denominator and set it equal to zero. So x + 4 = 0 -> x = -4.

    (Note that asymptotes are not numbers, but equations of lines. So to say that a vertical asymptote is -4 is not correct; you need to write x = -4.)

    To find the horizontal asymptote, first, note the degrees of the polynomials in the numerator (I'll denote d_n) and denominator ( d_d).

    If d_n < d_d , then the horizontal asymptote is y = 0.
    If d_n > d_d , then there is no horizontal asymptote. (There is a slant asymptote.)
    If d_n = d_d , then the horizontal asymptote is y = the ratio of the leading coefficients of the polynomials.

    In your problem, the degrees of the polynomials are the same (1). The leading coefficient of the polynomial in the numerator is -1, and the leading coefficient of the polynomial in the denominator is 1. So the horizontal asymptote is y = -1/1 = -1.


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  3. #3
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    Quote Originally Posted by s3a View Post
    How do I find the asymptotes of:

    y = -(x-8)/(x+4)

    I am assuming that it is (h,k) in f(x) = a/b(x-h) + k but there is no k meaning it should equal to zero however that is not one of the multiple choice answers for the horizontal asymptote. The vertical asymptote is the value of h, right? Meaning it should be - 4, right?

    Assuming that I am right for the vertical asymptote, someone please help me find the horizontal one.

    Any help would be GREATLY appreciated!
    Thanks!
    There are a number of ways to show that y = - \frac{x-8}{x + 4} = \frac{12}{x + 4} - 1 ....
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