A marathon of tension-filled reality shows can’t deliver the drama of quadratic functions. Peaks, valleys, hitting rock bottom – these plot twists do not exclusively belong to Hollywood. Quadratic functions depict the changing behavior of a set of data. Refer to the graph, *Pop Star’s Record Sales*.

Click **More Images** for a better view of the graph. Within 3 years after she inked her 1^{st} record deal, the singer had sold over 15 million records in 1 year. Two years later, the pop fizzled and she sold zero records. Algebra can depict a rags to riches to rags spectacle better than any melodrama.

### Characteristics of Quadratic Functions

1. Standard form is *y* = *ax*^{2 }+ *bx* + *c*, where *a≠* 0.

2. The graph is a **parabola**, a u-shaped figure.

3. The parabola will open upward or downward.

4. **A parabola that opens upward** contains a vertex that is a **minimum** point.

**A parabola that opens downward** contains a vertex that is a **maximum** point.

Click , to view *Parabola that opens upward* and *Parabola that opens downward*.

5. The **domain** of a quadratic function is all real numbers.

6. To determine the **range** of a quadratic function, ask yourself

two questions:

*Is the vertex a minimum or*

maximum?*What is the y-value of the vertex?*

If the vertex is a *minimum*, then the range is *all real numbers greater than or equal to the y-value*.

If the vertex is a *maximum*, then the range is *all real numbers less than or equal to the y-value*.

7. An **axis of symmetry** (also known as a **line of symmetry**) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form *x *= *n*, where *n* is a real number. Click **More Images** to view *Parabola that opens upward.* Its axis of symmetry is the vertical line *x* =0.

8. The *x***-intercepts** are the points at which a parabola intersects the *x*-axis. These points are also known as **zeroes**, **roots**, **solutions**, and **solution sets**. Each quadratic function will have two, one, or no *x*-intercepts.