1. **Stating the problem:** Create a similar problem to the given one: For $$y = \frac{2x - 1}{x^2 - x - 6}$$ find all asymptotes, state the domain and range, find the intercepts, determine increasing/decreasing intervals, and sketch the graph. 2. **New problem:** Consider the function $$y = \frac{3x + 2}{x^2 - 4x + 3}$$. 3. **Find all asymptotes:** - Vertical asymptotes occur where the denominator is zero: solve $$x^2 - 4x + 3 = 0$$. - Factor denominator: $$(x - 3)(x - 1) = 0$$ so vertical asymptotes at $$x = 1$$ and $$x = 3$$. - Horizontal asymptote: compare degrees of numerator and denominator (both degree 2 and 1 respectively). Since degree denominator > numerator, horizontal asymptote at $$y = 0$$. 4. **State the domain:** Domain is all real numbers except where denominator is zero: $$x \neq 1, 3$$. 5. **Find intercepts:** - x-intercepts: set numerator zero: $$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$$. - y-intercept: set $$x=0$$: $$y = \frac{3(0) + 2}{0 - 0 + 3} = \frac{2}{3}$$. 6. **Determine increasing/decreasing intervals:** - Find derivative $$y'$$ using quotient rule: $$y' = \frac{(3)(x^2 - 4x + 3) - (3x + 2)(2x - 4)}{(x^2 - 4x + 3)^2}$$ - Simplify numerator and find critical points by setting numerator zero. - Analyze sign of $$y'$$ to find intervals. 7. **Sketch the graph:** - Plot vertical asymptotes at $$x=1$$ and $$x=3$$. - Plot horizontal asymptote at $$y=0$$. - Plot intercepts at $$(-\frac{2}{3}, 0)$$ and $$(0, \frac{2}{3})$$. - Use increasing/decreasing intervals to shape the curve. This problem mirrors the original in structure and complexity.