Quadratic function

Definition

A quadratic function is a polynomial function of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola that opens upwards if $a > 0$ and downwards if $a < 0$.

5 Must Know Facts For Your Next Test

1. A quadratic function has the general form $f(x) = ax^2 + bx + c$.
2. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$.
3. The axis of symmetry is the vertical line given by $x = -\frac{b}{2a}$.
4. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the roots of the quadratic equation.
5. The discriminant, $\Delta = b^2 - 4ac$, determines the nature of the roots: two distinct real roots if $\Delta > 0$, one real root if $\Delta = 0$, and no real roots (two complex roots) if $\Delta < 0$.

Review Questions

• What is the standard form of a quadratic function?
• How do you find the vertex of a parabola given by a quadratic function?
• What does the discriminant tell you about the nature of the roots?

Related terms

Vertex: The highest or lowest point on the graph of a parabola, found at $(x, f(x))$ where $x = -\frac{b}{2a}$.

Axis of Symmetry: A vertical line that divides the parabola into two mirror images, given by $x = -\frac{b}{2a}$ for a quadratic function.

Discriminant: $\Delta = b^2 - 4ac$, used to determine the number and type of roots for a quadratic equation.