1. **Problem Statement:** Sketch the graph of the function \(y = \frac{6}{x+1}\) using transformations. Identify the domain, range, intercepts, and asymptotes. 2. **Formula and Important Rules:** The base function is \(y = \frac{1}{x}\), a hyperbola with vertical asymptote at \(x=0\) and horizontal asymptote at \(y=0\). Transformations: - Horizontal shift by \(-1\) (because of \(x+1\) in denominator). - Vertical stretch by factor 6. 3. **Domain:** The function is undefined where the denominator is zero: $$x+1=0 \implies x=-1$$ So, domain is \(\{x \mid x \neq -1, x \in \mathbb{R}\}\). 4. **Range:** Since the function is a vertical stretch of \(\frac{1}{x+1}\), the horizontal asymptote remains at \(y=0\), which the function never reaches. Thus, range is \(\{y \mid y \neq 0, y \in \mathbb{R}\}\). 5. **Intercepts:** - \(x\)-intercept: Set \(y=0\), solve \(\frac{6}{x+1}=0\). No solution since numerator 6 \(\neq 0\), so no \(x\)-intercept. - \(y\)-intercept: Set \(x=0\), then \(y=\frac{6}{0+1}=6\). 6. **Asymptotes:** - Vertical asymptote at \(x=-1\) (denominator zero). - Horizontal asymptote at \(y=0\) (limit as \(x \to \pm \infty\)). 7. **Summary:** - Domain: \(x \neq -1\) - Range: \(y \neq 0\) - \(x\)-intercept: none - \(y\)-intercept: 6 - Asymptotes: \(x=-1\), \(y=0\) This matches the given information and describes the graph as a hyperbola shifted left by 1 unit and vertically stretched by 6.