1. **State the problem:** We are given the rational expression $$\frac{2x - 3}{x + 4}$$ and want to understand its behavior and graph. 2. **Formula and rules:** This is a rational function of the form $$\frac{P(x)}{Q(x)}$$ where $$P(x) = 2x - 3$$ and $$Q(x) = x + 4$$. - The function is undefined where the denominator is zero, so find vertical asymptotes by solving $$x + 4 = 0$$. 3. **Find vertical asymptote:** $$x + 4 = 0 \implies x = -4$$ 4. **Find x-intercept:** Set numerator equal to zero: $$2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$$ 5. **Find y-intercept:** Set $$x=0$$: $$y = \frac{2(0) - 3}{0 + 4} = \frac{-3}{4}$$ 6. **Horizontal asymptote:** Since degrees of numerator and denominator are equal (both 1), horizontal asymptote is ratio of leading coefficients: $$y = \frac{2}{1} = 2$$ 7. **Summary:** - Vertical asymptote at $$x = -4$$ - Horizontal asymptote at $$y = 2$$ - x-intercept at $$x = \frac{3}{2}$$ - y-intercept at $$y = -\frac{3}{4}$$ This information helps sketch the graph and understand the function's behavior near asymptotes and intercepts.