1. **Problem Statement:** Given a quadratic function, factor it, find the x-intercepts, axis of symmetry, and vertex. 2. **General Form and Formula:** A quadratic function is generally written as $$y = ax^2 + bx + c$$. - The x-intercepts are the values of $x$ where $y=0$. - The axis of symmetry is the vertical line that passes through the vertex, given by $$x = -\frac{b}{2a}$$. - The vertex is the point $$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$. 3. **Example:** Let's pick the quadratic $$y = x^2 - 5x + 6$$. 4. **Factoring:** We look for two numbers that multiply to $6$ and add to $-5$. These are $-2$ and $-3$. $$y = (x - 2)(x - 3)$$ 5. **Finding x-intercepts:** Set $y=0$: $$0 = (x - 2)(x - 3)$$ So, $x - 2 = 0$ or $x - 3 = 0$ which gives $$x = 2 \quad \text{or} \quad x = 3$$ 6. **Axis of Symmetry:** Using the formula: $$x = -\frac{b}{2a} = -\frac{-5}{2 \times 1} = \frac{5}{2} = 2.5$$ 7. **Vertex:** Substitute $x=2.5$ into the original equation: $$y = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$ So the vertex is at: $$(2.5, -0.25)$$ **Final answers:** - Factored form: $$y = (x - 2)(x - 3)$$ - x-intercepts: $$x=2, 3$$ - Axis of symmetry: $$x=2.5$$ - Vertex: $$(2.5, -0.25)$$