To determine the vertical and slant asymptotes of the given rational function `f(x) = (48x^2-50x+9)/(-8x+3)`, begin by addressing the vertical asymptote. This involves setting the denominator equal to zero and solving for `x`, which pinpoints where the function is undefined. For the slant asymptote, applicable when the degree of the numerator exceeds that of the denominator by exactly one, perform polynomial long division. The result of this division, typically a linear expression, describes the end behavior of the function as `x` approaches infinity or negative infinity, effectively approximating the function's behavior at large magnitudes of `x`.