1. The problem is to find the axis of symmetry and the vertex of a quadratic function. 2. The axis of symmetry for a quadratic function $y = ax^2 + bx + c$ is given by the formula $$x = -\frac{b}{2a}$$. 3. Once the axis of symmetry $x$ is found, substitute it back into the quadratic equation to find the vertex's $y$-coordinate: $$y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$$. 4. Simplify the expression step-by-step: $$y = a\frac{b^2}{4a^2} - \frac{b^2}{2a} + c$$ $$y = \frac{b^2}{4a} - \frac{b^2}{2a} + c$$ 5. Use a common denominator to combine terms: $$y = \frac{b^2}{4a} - \frac{2b^2}{4a} + c = -\frac{b^2}{4a} + c$$ 6. Therefore, the vertex is at $$\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$. 7. This method helps find the vertex by first calculating the axis of symmetry and then substituting it back into the equation to get the vertex coordinates.