1. **State the problem:** We need to graph the rational function $$R(x) = \frac{x + 6}{x(x + 12)}$$. 2. **Identify intercepts:** - **x-intercepts:** Set numerator equal to zero: $$x + 6 = 0 \Rightarrow x = -6$$. - **y-intercept:** Set $$x=0$$, but denominator is zero at $$x=0$$, so no y-intercept. 3. **Find vertical asymptotes:** Set denominator equal to zero: $$x(x + 12) = 0 \Rightarrow x=0 \text{ or } x=-12$$. These are vertical asymptotes. 4. **Find horizontal asymptote:** Degree of numerator is 1, degree of denominator is 2, so horizontal asymptote is $$y=0$$. 5. **Analyze behavior near asymptotes:** - As $$x \to 0^+$$, denominator $$\to 0^+$$, numerator $$\to 6$$, so $$R(x) \to +\infty$$. - As $$x \to 0^-$$, denominator $$\to 0^-$$, numerator $$\to 6$$, so $$R(x) \to -\infty$$. - As $$x \to -12^+$$, denominator $$\to 0^-$$, numerator $$\to -6$$, so $$R(x) \to +\infty$$. - As $$x \to -12^-$$, denominator $$\to 0^+$$, numerator $$\to -6$$, so $$R(x) \to -\infty$$. 6. **Summary:** - Vertical asymptotes at $$x=0$$ and $$x=-12$$. - x-intercept at $$x=-6$$. - Horizontal asymptote at $$y=0$$. - Behavior near asymptotes as above. 7. **Graphing:** Plot vertical lines at $$x=0$$ and $$x=-12$$. Plot point at $$(-6,0)$$. Draw curve approaching asymptotes and intercepts respecting the behavior. Final answer: The graph of $$R(x) = \frac{x + 6}{x(x + 12)}$$ has vertical asymptotes at $$x=0$$ and $$x=-12$$, an x-intercept at $$x=-6$$, and a horizontal asymptote at $$y=0$$, with the described behavior near asymptotes.