key term - Quadratic function

Definition

A quadratic function is a polynomial function of degree 2, typically expressed in the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants and $a \neq 0$. The graph of a quadratic function is a parabola that opens either upwards or downwards.

5 Must Know Facts For Your Next Test

1. The vertex of a quadratic function given by $f(x) = ax^2 + bx + c$ can be found using the formula $(-\frac{b}{2a}, f(-\frac{b}{2a}))$.
2. The axis of symmetry of the parabola is the vertical line $x = -\frac{b}{2a}$.
3. The direction in which the parabola opens is determined by the sign of coefficient $a$: it opens upwards if $a > 0$ and downwards if $a < 0$.
4. The roots (or zeros) of the quadratic function can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
5. The discriminant, given by $b^2 - 4ac$, determines the nature of the roots: if it's positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.

Review Questions

Related terms

Vertex:

The highest or lowest point on the graph of a quadratic function, located at ($-\frac{b}{2a}$, $f(-\frac{b}{2a})$).

Axis of Symmetry:

A vertical line that divides the parabola into two mirror images, represented by $x = -\frac{b}{2a}$.

Discriminant:

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