Functions

1. Overview and Algebraic Forms

A quadratic function is a second-degree polynomial function. The general representations include:

• Standard Form:

  • $f(x) = ax^2 + bx + c$

  • Here, $a$, $b$, and $c$ are real constants with $a \neq 0$.

• Vertex Form:

  • $f(x) = a(x - h)^2 + k$

  • The point $(h, k)$ represents the vertex of the parabola.

• Factored (Intercept) Form:

  • $f(x) = a(x - r<em>1)(x - r</em>2)$

  • The values $r<em>1$ and $r</em>2$ represent the x-intercepts (roots) of the function.

2. Key Features of Parabolas

• Direction of Opening:

  • If $a > 0$, the parabola opens upward, creating a minimum point at the vertex.

  • If $a < 0$, the parabola opens downward, creating a maximum point at the vertex.

• Axis of Symmetry:

  • A vertical line given by $x = -\frac{b}{2a}$.

• Vertex Coordinate:

  • Calculated as $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.

• y-Intercept:

  • The point where the graph crosses the y-axis, located at $(0, c)$.

3. Methods for Solving Quadratic Equations

To find the roots where $f(x) = 0$:

1. Factoring:

   • Rewrite $ax^2 + bx + c = 0$ into linear factors $(px + q)(rx + s) = 0$ and set each factor equal to zero.

2. Completing the Square:

   • Transform $ax^2 + bx + c = 0$ into the form $(x - h)^2 = d$ to solve directly by taking square roots.

3. Quadratic Formula:

   • Applicable to any quadratic equation:

     • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

4. The Discriminant

The discriminant determines the nature and number of roots of a quadratic equation:

• Formula:

  • $D = b^2 - 4ac$

• Root Conditions:

  • If $D > 0$: Two distinct real solutions.

  • If $D = 0$: One repeated real solution.

  • If $D < 0$: Two complex conjugate solutions.