To represent a rational function with these conditions, we'll start by representing the reciprocal of a quadratic function, which has the form 1/(ax^2 + bx + c).
Next, let's consider the asymptotes. We have three asymptotes: x = -5, x = 4/3, and y = 0. For the vertical asymptotes, we need factors that will make the denominator equal to zero at x = -5 and x = 4/3. Thus, the denominator will be of the form (x - (-5))(x - (4/3)) = (x + 5)(x - 4/3).
For the horizontal asymptote, since y = 0, we know that the degree of the numerator is less than or equal to the degree of the denominator. This means that the quadratic in the denominator must have a higher degree than the numerator.
Considering the y-intercept of 1.5, we can set x = 0 in our equation to find the value of the numerator. Since the denominator has factors of (x + 5) and (x - 4/3), the y-intercept occurs when x = 0, so we can write the numerator as k(x - 0) = kx. Since the y-intercept is 1.5, we can set up the equation as follows: 1.5 = k(0) = 0, which means k = 1.5.
Putting everything together, our rational function equation is:
f(x) = (1.5x)/((x + 5)(x - 4/3))
This equation satisfies the conditions of being the reciprocal of a quadratic function, having the given asymptotes, y < 0 whenever x < -5, and having a y-intercept of 1.5.