General form of a quadratic function

Definition

The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$. This representation is crucial for solving quadratic equations and analyzing their properties.

5 Must Know Facts For Your Next Test

1. In the general form, the coefficient $a$ determines the parabola's direction (upwards if $a > 0$, downwards if $a < 0$).
2. The discriminant, given by $b^2 - 4ac$, helps determine the nature of the roots (real and distinct, real and equal, or complex).
3. The vertex of the parabola can be found using the formula $\left(\frac{-b}{2a}, \frac{-D}{4a}\right)$ where $D = b^2 - 4ac$.
4. Completing the square or using the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$) are methods to solve for the roots.
5. The axis of symmetry of a quadratic function in general form is given by the line $x = \frac{-b}{2a}$.

Review Questions

Related terms

Vertex Form:

A form of a quadratic function that makes it easy to identify its vertex: $y = a(x - h)^2 + k$.

Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, used to find solutions to any quadratic equation.

Discriminant:

$D = b^2 - 4ac$, used to determine the nature and number of roots of a quadratic equation.