1. The problem is to analyze and understand the graph of the rational function $$y = \frac{3}{x - 4} - 6$$. 2. The general form of a rational function with a vertical asymptote is $$y = \frac{a}{x - h} + k$$ where: - $x = h$ is the vertical asymptote (where the denominator is zero). - $y = k$ is the horizontal asymptote (the value the function approaches as $x \to \pm \infty$). 3. For the given function, the denominator is $x - 4$, so the vertical asymptote is at: $$x = 4$$ 4. The horizontal asymptote is the constant term outside the fraction, which is: $$y = -6$$ 5. The function shifts the basic hyperbola $y = \frac{3}{x}$ right by 4 units and down by 6 units. 6. The numerator 3 is positive, so the branches of the hyperbola will be: - To the left of $x=4$, the function values are negative and approach $-6$ from below. - To the right of $x=4$, the function values are positive and approach $-6$ from above. 7. This matches the description: red dashed asymptotes at $x=4$ and $y=-6$, with the left branch below $y=-6$ and the right branch above $y=-6$. Final answer: The graph has vertical asymptote $x=4$, horizontal asymptote $y=-6$, and the hyperbola branches as described.