1. **State the problem:** Find the horizontal asymptote, vertical asymptote, x-intercept, y-intercept, and hole of the function $$f(x) = \frac{3x - 7}{4x + 14}$$. 2. **Horizontal asymptote:** For rational functions $$\frac{P(x)}{Q(x)}$$ where degrees of numerator and denominator are equal, horizontal asymptote is $$y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$$. Here, degree numerator = degree denominator = 1. Leading coefficient numerator = 3. Leading coefficient denominator = 4. So, horizontal asymptote is $$y = \frac{3}{4}$$. 3. **Vertical asymptote:** Set denominator equal to zero and solve for $$x$$: $$4x + 14 = 0$$ $$4x = -14$$ $$x = \cancel{\frac{4x}{4}}\frac{-14}{\cancel{4}} = -\frac{14}{4} = -\frac{7}{2}$$ So, vertical asymptote is $$x = -\frac{7}{2}$$. 4. **x-intercept:** Set numerator equal to zero and solve for $$x$$: $$3x - 7 = 0$$ $$3x = 7$$ $$x = \cancel{\frac{3x}{3}}\frac{7}{\cancel{3}} = \frac{7}{3}$$ So, x-intercept is $$\left(\frac{7}{3}, 0\right)$$. 5. **y-intercept:** Evaluate $$f(0)$$: $$f(0) = \frac{3(0) - 7}{4(0) + 14} = \frac{-7}{14} = -\frac{1}{2}$$ So, y-intercept is $$(0, -\frac{1}{2})$$. 6. **Hole:** Check if numerator and denominator have common factors. Factor denominator: $$4x + 14 = 2(2x + 7)$$ Numerator: $$3x - 7$$ cannot be factored further. No common factors, so no hole. **Final answers:** - Horizontal asymptote: $$y = \frac{3}{4}$$ - Vertical asymptote: $$x = -\frac{7}{2}$$ - x-intercept: $$\left(\frac{7}{3}, 0\right)$$ - y-intercept: $$(0, -\frac{1}{2})$$ - Hole: None