1. **Problem Statement:** We are studying the quadratic function in the form $$f(x) = ax^2 + bx + c$$ where $a,b,c \in \mathbb{R}$ and $a \neq 0$. 2. **Completing the Square Method:** The quadratic function can be rewritten in vertex form by completing the square: $$f(x) = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b^2 - 4ac}{4a^2}\right) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}$$ 3. **Interpretation of Terms:** - The term $\left(x + \frac{b}{2a}\right)^2$ represents the squared binomial. - The constant term $-\frac{b^2 - 4ac}{4a}$ shifts the parabola vertically. - The vertex of the parabola is at $$\left(-\frac{b}{2a}, -\frac{b^2 - 4ac}{4a}\right)$$. 4. **Graph Shape and Vertex:** - If $a > 0$, the parabola opens upwards and the vertex is a minimum point. - If $a < 0$, the parabola opens downwards and the vertex is a maximum point. 5. **Discriminant and Graph Behavior:** - When $b^2 - 4ac < 0$, the parabola does not intersect the x-axis. - If $a > 0$, $f(x) > 0$ for all $x$. - If $a < 0$, $f(x) < 0$ for all $x$. - When $b^2 - 4ac = 0$, the parabola touches the x-axis at one point (vertex on x-axis). - When $b^2 - 4ac > 0$, the parabola intersects the x-axis at two distinct points. 6. **Factoring Using Roots:** If $b^2 - 4ac > 0$, let $$k = \frac{\sqrt{b^2 - 4ac}}{2a}$$ and $$\lambda_1 = \frac{b}{2a} - k, \quad \lambda_2 = \frac{b}{2a} + k$$ then $$f(x) = a(x + \lambda_1)(x + \lambda_2)$$ 7. **Domain and Range:** - The domain of any quadratic function is all real numbers: $$(-\infty, \infty)$$. - The range depends on $a$ and the vertex: - If $a > 0$, range is $$\left[-\frac{b^2 - 4ac}{4a}, \infty\right)$$. - If $a < 0$, range is $$\left(-\infty, -\frac{b^2 - 4ac}{4a}\right]$$. 8. **Intercepts:** - The y-intercept is at $f(0) = c$. - The x-intercepts depend on the roots and discriminant. **Summary:** The quadratic function $f(x) = ax^2 + bx + c$ can be expressed in vertex form by completing the square, revealing the vertex and graph shape. The discriminant $b^2 - 4ac$ determines the nature of the roots and the position of the parabola relative to the x-axis. The parabola opens upwards if $a > 0$ and downwards if $a < 0$. The domain is all real numbers, and the range depends on the vertex and $a$.