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Design a rational function with the following characteristics: three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes x=-2 and x=3; oblique asymptote y=2x+1.

  • Math -

    Well first lets get one with three real zeros.

    y = k(x-x1)(x-x2)(x-x3)
    but one happens twice
    y = k(x-x1)(x-x2)(x-x2)
    denominator = 0 at x=-2 and x = 3
    y = c[(x-x1)(x-x2)(x-x2)]/[(x+2)(x-3)]
    when x = 0, y = 1
    y =1 = c[-x1 x2^2]/-6
    6 = c x1 x2^2
    when x and y are big, y = 2x
    big x
    y = c x^3/x^2 = c x
    so c = 2
    so
    x1 x2^2 = 3 or x1 = x2^2/3
    y = 2[(x-x1)(x-x2)(x-x2)]/[(x+2)(x-3)]
    y = 2[(x-x2^2/3)(x-x2)(x-x2)]/[(x+2)(x-3)]
    hmmm, unless I am missing something I am still free to choose x2

  • error -

    x1 x2^2 = 3 or x1 = 3/x2^2
    y = 2[(x-x1)(x-x2)(x-x2)]/[(x+2)(x-3)]
    y = 2[(x - 3/x2^2)(x-x2)(x-x2)]/[(x+2)(x-3)]

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