1. **Stating the problem:** We are given the function $$y = \frac{1}{x} + 5$$ and asked to understand and explain its graphs and asymptotes. 2. **Analyze the function:** The function is a transformation of the basic reciprocal function $$y = \frac{1}{x}$$. 3. **Vertical asymptote:** The term $$\frac{1}{x}$$ is undefined at $$x=0$$, so there is a vertical asymptote at $$x=0$$. 4. **Horizontal asymptote:** As $$x \rightarrow \pm \infty$$, $$\frac{1}{x} \rightarrow 0$$, so $$y \rightarrow 5$$. This means there is a horizontal asymptote at $$y=5$$. 5. **Graph behavior:** The graph looks like a hyperbola shifted 5 units up from the parent function $$y=\frac{1}{x}$$. 6. **Explanation of the image:** - The bottom-right graph matches $$y=\frac{1}{x}+5$$ because it has a vertical asymptote at $$x=0$$ (red dashed line) and the horizontal asymptote near $$y=5$$ (curve approaches 5). - Other graphs have vertical asymptotes at different $$x$$ values, so they represent functions like $$\frac{1}{x - a}$$ with shifts in $$x$$, which are different from $$y=\frac{1}{x}+5$$. **Final answer:** The function $$y = \frac{1}{x} + 5$$ has a vertical asymptote at $$x=0$$ and a horizontal asymptote at $$y=5$$, shifting the original reciprocal function 5 units up. The bottom-right graph corresponds to this function.