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Math Help - Rational Function question

  1. #1
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    Rational Function question

    Given the rational function f(x) = 2x^3 - 7x + 3 / x^2 - 4

    a) Determine all of the asymptotes of f(x). Explain your work clearly.
    b) Make a neat sketch of f(x). Label all important parts.

    I determined that the vertical asymptotes are x = -2, 2 (hopefully I'm right), and there are no horizontal asymptotes. I'm wondering if there is a slant asymptote? How can I tell? If there is one how do I find what the slant asymptote is?

    How would the graph look like?

    Thanks for your time.
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  2. #2
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    Quote Originally Posted by zuuberbat View Post
    Given the rational function f(x) = 2x^3 - 7x + 3 / x^2 - 4

    ...
    I'm wondering if there is a slant asymptote? How can I tell? If there is one how do I find what the slant asymptote is?

    How would the graph look like?

    Thanks for your time.
    I assume that you mean:

    f(x) = \dfrac{2x^3-7x+3}{x^2-4}

    If so: Do the long division to rearrange the term of the function:

    (2x^3-7x+3) \div (x^2-4) = 2x+\dfrac{x+3}{x^2-4}

    If |x| approaches infinity the value of the fraction is going to be zero. Therefore the asymptote has the equation:

     y= 2x
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