Quadratic Form 4D92E6

1. **State the problem:** We are given a quadratic function $$y = a x^2 + b x + c$$ where $$a$$, $$b$$, and $$c$$ are constants. 2. **Formula and explanation:** This is the general form of a quadratic equation. The graph of this function is a parabola. 3. **Important rules:** - If $$a > 0$$, the parabola opens upwards. - If $$a < 0$$, the parabola opens downwards. - The vertex of the parabola is at $$x = -\frac{b}{2a}$$. - The axis of symmetry is the vertical line $$x = -\frac{b}{2a}$$. 4. **Intermediate work:** - To find the vertex, substitute $$x = -\frac{b}{2a}$$ into the equation: $$y = a \left(-\frac{b}{2a}\right)^2 + b \left(-\frac{b}{2a}\right) + c$$ - Simplify: $$y = a \frac{b^2}{4a^2} - \frac{b^2}{2a} + c = \frac{b^2}{4a} - \frac{b^2}{2a} + c$$ - Combine terms: $$y = -\frac{b^2}{4a} + c$$ 5. **Final vertex coordinates:** $$\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$ This vertex represents the maximum or minimum point of the parabola depending on the sign of $$a$$. 6. **Summary:** The quadratic function $$y = a x^2 + b x + c$$ describes a parabola with vertex at $$\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$ and axis of symmetry $$x = -\frac{b}{2a}$$.