math(precalculus)

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Write the equation of a Rational Function satisfying given conditions.

Has vertical asymptotes located at x-2 and x=-1
Has a horizontal asymptote located at y=0(x-axis)
y-intercept:(0,2) x-intercept (4,0)

R(x)= ???

what I did is,
R(x)= ax+b/c(x-2)(x+1)
then plug in (0,2), b=-4c
plug in(4,0), a=c
then what to do next?? Please help me with this I will appreciate that!!

  • math(precalculus) -

    You don't need any c. It can be absorbed into a and b.

    Actually, you have solved the problem. Pick any value for c, and plug it in. As long as a=c and b=-4c, R(x) will work. So, make things easy. Let c=1.

    As I worked it out,

    R(x) = ???/(x+1)(x-2)

    Since y=0 is the asymptote, you know that the degree of the numerator is less than the denominator. So,

    R(x) = (ax+b) / (x+1)(x-2)
    Now, with the intercepts, you know that

    b/(1)(-2) = 2, so b = -4
    a(4)+b = 0, so 4a-4=0, so a=1

    R(x) = (x-4)/(x+1)(x-2)

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