1. **State the problem:** Convert a quadratic function from general form $y = ax^2 + bx + c$ to standard (vertex) form $y = a(x-h)^2 + k$. 2. **Formula and rules:** The vertex form reveals the vertex $(h,k)$ of the parabola. To convert, we complete the square. 3. **Step-by-step process:** - Start with $y = ax^2 + bx + c$. - Factor out $a$ from the first two terms: $y = a(x^2 + \frac{b}{a}x) + c$. - Complete the square inside the parentheses: add and subtract $\left(\frac{b}{2a}\right)^2$. $$y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$ - Rewrite as: $$y = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c$$ - Distribute $a$: $$y = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$ - Combine constants: $$y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$$ 4. **Interpretation:** The vertex is at $\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$. This is the standard form $y = a(x-h)^2 + k$ with $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$. This method helps identify the vertex and graph the parabola easily.