Math

Write an equation to represent a rational function with the following conditions:
- reciprocal of a quadratic function
- asymptotes with the equations x = -5, x= 4/3, and y = 0
- y < 0 whenever x < -5
- y-intercept of 1.5

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  1. the roots are -5 and 4/3, so start with
    y = a/((x+5)(3x-4))
    This also has y=0 as the horizontal asymptote
    When x < -5, both factors are negative, so a must be negative
    Now, using the point (0,3/2), we get
    3/2 = a/(-5 * -19)
    a = 142.5
    But that is positive. So we need to make that denominator positive, without changing the asymptotes. One way is to square a factor.
    y = a/((x+5)^2 (3x-4))
    Now at x=0,
    3/2 = a/(25 * -4)
    a = -150
    and so
    y = -150/((x+5)^2 (3x-4)) = -150/(3x^3+26x^2+35x-100)

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