1. **State the problem:** Graph the function $$f(x) = \frac{4}{x-5}$$ and identify its transformations and asymptotes. 2. **Recall the general form and asymptotes:** The parent function for inverse variation is $$f(x) = \frac{1}{x}$$. - Vertical asymptote (V.A.) occurs where the denominator is zero. - Horizontal asymptote (H.A.) is the value that $$f(x)$$ approaches as $$x \to \pm \infty$$. 3. **Identify transformations:** - The function is shifted right by 5 units because of $$x-5$$ in the denominator. - The numerator 4 indicates a vertical stretch by a factor of 4. 4. **Find asymptotes:** - Vertical asymptote: Set denominator to zero $$x-5=0 \Rightarrow x=5$$. - Horizontal asymptote: Since degree of numerator is 0 and denominator is 1, $$y=0$$. 5. **Graph behavior:** - For $$x > 5$$, $$f(x)$$ is positive and approaches 0 from above as $$x \to \infty$$. - For $$x < 5$$, $$f(x)$$ is negative and approaches 0 from below as $$x \to -\infty$$. 6. **Summary:** - Transformations: Shift right 5 units, vertical stretch by 4. - Vertical asymptote: $$x=5$$. - Horizontal asymptote: $$y=0$$. Final answer: The graph of $$f(x) = \frac{4}{x-5}$$ has a vertical asymptote at $$x=5$$ and a horizontal asymptote at $$y=0$$, shifted right 5 units with vertical stretch 4.