Quadratic function

Definition

A quadratic function is a polynomial function of degree 2, typically in the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. 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. The vertex of the quadratic function $f(x) = ax^2 + bx + c$ can be found at the point $(\frac{-b}{2a}, f(\frac{-b}{2a}))$.
2. The axis of symmetry for the parabola represented by a quadratic function is the vertical line $x = \frac{-b}{2a}$.
3. The roots or zeros of the quadratic function are found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4. The discriminant, given by $\Delta = b^2 - 4ac$, determines the nature of the roots: two real and distinct roots if $\Delta > 0$, one real repeated root if $\Delta = 0$, and two complex roots if $\Delta < 0$.
5. Completing the square is a method to rewrite a quadratic function in vertex form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Review Questions

• What is the general form of a quadratic function?
• How do you find the vertex and axis of symmetry for a given quadratic function?
• Explain how to determine the nature of the roots using the discriminant.

Related terms

Parabola: A symmetric curve formed by all points equidistant from a point (focus) and a line (directrix). It represents the graph of a quadratic function.

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

Vertex Form: $f(x) = a(x-h)^2 + k$: A way to express a quadratic function highlighting its vertex $(h, k)$.