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Math Help - Rational Functions

  1. #1
    Junior Member
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    Wink Rational Functions

    Please Help:

    Write an equation of a rational function of the form:

    f(x) = ax + b / cx + d;

    whose graph has the following feautures:

    x-intercept of 1/4

    y-intercept of -1/2

    vertical asymptote w/equation x = -2/3

    horizontal asymptote w/equation y = 4/3


    Thank you
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by advancedfunctions2010 View Post
    Please Help:

    Write an equation of a rational function of the form:

    f(x) = ax + b / cx + d;

    whose graph has the following feautures:

    x-intercept of 1/4

    y-intercept of -1/2

    vertical asymptote w/equation x = -2/3

    horizontal asymptote w/equation y = 4/3


    Thank you
    I suspect that you are asked to find a function of the form f(x)=(ax+b)/(cx+d) and that you were just a bit lazy in leaving out the paranthesis around the denominator and numerator terms.
    If so, the following system of equations can be used to determine a,b,c, and d:

    \begin{array}{lcrcl}<br />
\text{(1)}&\qquad & a\cdot\tfrac{1}{4}+b &=& 0\\[.2cm]<br />
\text{(2)} && \tfrac{b}{d} &=& -\tfrac{1}{2}\\[.2cm]<br />
\text{(3)} && c\cdot\left(-\tfrac{2}{3}\right) +d &=& 0\\[.2cm]<br />
\text{(4)} && \tfrac{a}{c} &=& \tfrac{4}{3}<br />
\end{array}
    This system has infinitenly many solutions, because it is possible to multiply numerator and denominator with the same constant \neq 0 to get another solution of the original problem. One, particularly nice solution (i.e. one with small, integral values for a,b,c, and d) is a=4, b=-1, c=3, d=2.
    Last edited by Failure; April 7th 2010 at 12:09 PM.
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  3. #3
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    thank you
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  4. #4
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    Grandad's Avatar
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    Hello everyone

    Just a quick correction to a typo. I'm sure Failure intended equation (1) to read:
    a\cdot\tfrac14+b=0
    Grandad
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