1. **Problem:** Solve and analyze the quadratic function $$y = x^2 + 5x + 6$$. Step 1: Factor the quadratic. $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ Step 2: Find the roots by setting each factor to zero. $$x + 2 = 0 \Rightarrow x = -2$$ $$x + 3 = 0 \Rightarrow x = -3$$ Step 3: Find the vertex using $$x = -\frac{b}{2a} = -\frac{5}{2} = -2.5$$. Step 4: Calculate $$y$$ at vertex: $$y = (-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$ Step 5: The parabola opens upward (since $$a=1>0$$), vertex at $$(-2.5, -0.25)$$. 2. **Problem:** Solve and analyze $$y = 2x^2 - 3x - 1$$. Step 1: Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{9 + 8}}{4} = \frac{3 \pm \sqrt{17}}{4}$$ Step 2: Calculate approximate roots: $$x_1 \approx \frac{3 + 4.123}{4} = 1.78$$ $$x_2 \approx \frac{3 - 4.123}{4} = -0.28$$ Step 3: Vertex at $$x = -\frac{b}{2a} = \frac{3}{4} = 0.75$$. Step 4: Calculate $$y$$ at vertex: $$y = 2(0.75)^2 - 3(0.75) - 1 = 2(0.5625) - 2.25 - 1 = 1.125 - 3.25 = -2.125$$ Step 5: Parabola opens upward (since $$a=2>0$$), vertex at $$(0.75, -2.125)$$. 3. **Problem:** Solve and analyze $$y = x^2 - 4x - 5$$. Step 1: Factor: $$x^2 - 4x - 5 = (x - 5)(x + 1)$$ Step 2: Roots: $$x = 5, -1$$ Step 3: Vertex at $$x = -\frac{b}{2a} = 2$$. Step 4: Calculate $$y$$ at vertex: $$y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9$$ Step 5: Parabola opens upward, vertex at $$(2, -9)$$. 4. **Problem:** Analyze the linear function $$y = 5x + 6$$. Step 1: This is a straight line with slope 5 and y-intercept 6. Step 2: Find x-intercept by setting $$y=0$$: $$0 = 5x + 6 \Rightarrow x = -\frac{6}{5} = -1.2$$ 5. **Problem:** Analyze the linear function $$y = 2x + \frac{6}{2}$$. Step 1: Simplify constant term: $$\frac{6}{2} = 3$$ Step 2: So, $$y = 2x + 3$$. Step 3: Slope is 2, y-intercept is 3. Step 4: Find x-intercept: $$0 = 2x + 3 \Rightarrow x = -\frac{3}{2} = -1.5$$ **Final answers:** 1. Roots: $$-3, -2$$; Vertex: $$(-2.5, -0.25)$$ 2. Roots: $$\frac{3 \pm \sqrt{17}}{4}$$ approx $$1.78, -0.28$$; Vertex: $$(0.75, -2.125)$$ 3. Roots: $$5, -1$$; Vertex: $$(2, -9)$$ 4. Line with slope 5, y-intercept 6, x-intercept $$-1.2$$ 5. Line with slope 2, y-intercept 3, x-intercept $$-1.5$$