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Math Help - Functions and Curve Sketching

  1. #1
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    Functions and Curve Sketching

    A curve is given by the equation y = \frac{ax^2 + bx + c}{dx^2 + ex + f} . The equations of the asymptotes to the curve are x = 1, x = 3 and y = 1. The curve passes through the points (2,0) and the origin..

    I figured how to find d, e and f by taking the x asymptotes and make them = 0 ...getting \frac{ax^2 + bx + c}{(x-1)(x-3)} I can't seem to figure out how to find a b and c though..any help would be greatly appreciated.

    Thanks in advance
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  2. #2
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    Quote Originally Posted by kleyzam View Post
    A curve is given by the equation y = \frac{ax^2 + bx + c}{dx^2 + ex + f} . The equations of the asymptotes to the curve are x = 1, x = 3 and y = 1. The curve passes through the points (2,0) and the origin..

    I figured how to find d, e and f by taking the x asymptotes and make them = 0 ...getting \frac{ax^2 + bx + c}{(x-1)(x-3)} I can't seem to figure out how to find a b and c though..any help would be greatly appreciated.

    Thanks in advance
    1. Since f(0) = 0 follows c = 0

    2. If \lim_{|x|\rightarrow \infty}\left(\dfrac{ax^2 + bx}{x^2-4x+3}  \right)=1~\implies~ a = 1

    3. If f(2) = 0 then 2^2+2b=0~\implies~b = -2

    4. Therefore f(x)=\dfrac{x^2 -2x}{x^2-4x+3}
    Attached Thumbnails Attached Thumbnails Functions and Curve Sketching-drei_asympt.png  
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