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What Are Quadratic Functions?

Dramatic Quadratic

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Pop Star's Record Sales

Pop Star's Record Sales

Jennifer Ledwith

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 1st 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+ 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.

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