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 $$x = -\frac{b}{2a}$$ where $$a=1$$ and $$b=5$$. $$x = -\frac{5}{2 \times 1} = -\frac{5}{2} = -2.5$$ Step 4: Calculate $$y$$ at vertex. $$y = (-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$ Step 5: Summary: - Roots: $$x = -3, -2$$ - Vertex: $$(-2.5, -0.25)$$ - Parabola opens upward since $$a=1 > 0$$. 2. **Problem:** Solve and analyze $$y = 2x^2 - 3x - 1$$. Step 1: Use quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $$a=2$$, $$b=-3$$, $$c=-1$$. Step 2: Calculate discriminant. $$\Delta = (-3)^2 - 4 \times 2 \times (-1) = 9 + 8 = 17$$ Step 3: Calculate roots. $$x = \frac{3 \pm \sqrt{17}}{4}$$ Step 4: Find vertex. $$x = -\frac{b}{2a} = -\frac{-3}{4} = \frac{3}{4} = 0.75$$ Step 5: 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 6: Summary: - Roots: $$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$$ - Vertex: $$(0.75, -2.125)$$ - Parabola opens upward since $$a=2 > 0$$. 3. **Problem:** Solve and analyze $$y = x^2 - 4x - 5$$. Step 1: Factor the quadratic. $$x^2 - 4x - 5 = (x - 5)(x + 1)$$ Step 2: Find roots. $$x - 5 = 0 \Rightarrow x = 5$$ $$x + 1 = 0 \Rightarrow x = -1$$ Step 3: Find vertex. $$x = -\frac{b}{2a} = -\frac{-4}{2} = 2$$ Step 4: Calculate $$y$$ at vertex. $$y = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9$$ Step 5: Summary: - Roots: $$x = 5, -1$$ - Vertex: $$(2, -9)$$ - Parabola opens upward since $$a=1 > 0$$. 4. **Problem:** Analyze the linear function $$y = 5x + 6$$. Step 1: Identify slope and y-intercept. - Slope $$m = 5$$ - Y-intercept $$b = 6$$ Step 2: Find x-intercept by setting $$y=0$$. $$0 = 5x + 6 \Rightarrow x = -\frac{6}{5} = -1.2$$ Step 3: Summary: - Slope: 5 - Y-intercept: 6 - X-intercept: -1.2 5. **Problem:** Analyze the linear function $$y = 2x + \frac{6}{2}$$. Step 1: Simplify constant term. $$\frac{6}{2} = 3$$ Step 2: Rewrite function. $$y = 2x + 3$$ Step 3: Find x-intercept. $$0 = 2x + 3 \Rightarrow x = -\frac{3}{2} = -1.5$$ Step 4: Summary: - Slope: 2 - Y-intercept: 3 - X-intercept: -1.5 **Final summary:** - Quadratic functions (1, 2, 3) have roots and vertices as calculated. - Linear functions (4, 5) have slopes and intercepts as above. These can be graphed on a coordinate plane showing parabolas opening upwards for quadratics and straight lines for linear functions.