1. **Problem 1:** Given the function $$h = \frac{3s + 1}{1 - 2s}$$ Find the vertical and horizontal asymptotes. 2. **Step 1:** Vertical asymptote occurs where the denominator is zero. Set $$1 - 2s = 0$$ Solve for $$s$$: $$2s = 1 \implies s = \frac{1}{2}$$ So, vertical asymptote is $$s = \frac{1}{2}$$. 3. **Step 2:** Horizontal asymptote is found by comparing degrees of numerator and denominator. Both numerator and denominator are degree 1. Horizontal asymptote is $$y = \frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}} = \frac{3}{-2} = -\frac{3}{2}$$. 4. **Problem 2:** Given the function $$f = \frac{3 + 2s - 4s - 3s - 1}{1 - 3s}$$ Simplify numerator: $$3 + 2s - 4s - 3s - 1 = (3 - 1) + (2s - 4s - 3s) = 2 - 5s$$ So, $$f = \frac{2 - 5s}{1 - 3s}$$ Compare with values 1.25, 4.05, 0.65, 0.50 (likely function values or parameters). 5. **Problem 3:** Given $$\frac{6 + s - (\text{vertical} - 3s)}{9 - 2s} = 8$$ Assuming vertical is a constant or expression, rewrite numerator: $$6 + s - \text{vertical} + 3s = (6 - \text{vertical}) + 4s$$ Set the whole expression equal to 8: $$\frac{(6 - \text{vertical}) + 4s}{9 - 2s} = 8$$ Multiply both sides: $$(6 - \text{vertical}) + 4s = 8(9 - 2s) = 72 - 16s$$ Bring all terms to one side: $$(6 - \text{vertical}) + 4s - 72 + 16s = 0 \implies (6 - 72 - \text{vertical}) + (4s + 16s) = 0$$ Simplify: $$(-66 - \text{vertical}) + 20s = 0$$ Solve for vertical: $$\text{vertical} = -66 + 20s$$ Given comparison values 5 ± 2, 4 ± 1, 5 ± 6. 6. **Problem 4:** Given $$\frac{(3s + 7)(s^2 + 5)}{3s + 8 + \text{expression}} = 2$$ Expression in denominator is complex: $$6 \quad 4.5 \quad 35 \quad 253 + 3s^a - (2 + 3) \text{something} - 1$$ Denominator: $$1 - 20 s^a$$ Graph points given: (-2, -20), (-1, -0.5), (0, 5), (2, 5) **Summary:** - Problem 1 vertical asymptote: $$s = \frac{1}{2}$$ - Problem 1 horizontal asymptote: $$y = -\frac{3}{2}$$ - Problem 2 simplified function: $$f = \frac{2 - 5s}{1 - 3s}$$ - Problem 3 vertical expression: $$\text{vertical} = -66 + 20s$$ - Problem 4 complex rational function with given points.