By Kardi Teknomo, PhD.

AHP

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Quadratic Function

First, let us distinguish several similar terms. What is the different between quadratic function from quadratic equation and quadratic formula?

Function Quadratic Function is called quadratic function when the three parameters Quadratic Function , Quadratic Function and Quadratic Function are real numbers. Quadratic function is more general function than a linear function because when the value of parameter Quadratic Function , then the quadratic function degenerate becomes a linear function Quadratic Function (linear means straight line geometrically). When the function value is set to be equal to zero (i.e. Quadratic Function ), the quadratic function is called quadratic equation

Quadratic Function .

The formula to find the roots of quadratic equation is called quadratic formula

Quadratic Function

or often called ABC formula because it involves three parameters Quadratic Function , Quadratic Function and Quadratic Function .

We will explore more the three quadratic terms above. First, we discuss about the characteristics of quadratic function.

Geometrically, quadratic function Quadratic Function produces a parabola curve when we plot in Quadratic Function plane. The shape and characteristics of the quadratic function is closely related to the parabola and depends on the value of the parameters.

Quadratic Function

Quadratic Function

Smiling parabola when Quadratic Function

Sad parabola when Quadratic Function

  • The parameter attached to the quadratic term ( Quadratic Function ) is called quadratic parameter or shape parameter because it influences the shape of the parabola. When the quadratic parameter is positive ( Quadratic Function ), the parabola is opened upward, look like a smile (it is sometimes called concave up ). The lowest or highest point on the parabola is called extreme point . For a smiling parabola, the extreme point is at minimum.
  • When the quadratic parameter is negative ( Quadratic Function ), the parabola is opened downward, look like a sad mouth (it is sometimes called concave down ). The extreme point of sad parabola is at maximum.
  • Of course, if the quadratic parameter is zero ( Quadratic Function ), the quadratic function becomes linear function and has no extreme point.
  • The second parameter ( Quadratic Function ) is the gradient of linear function degenerated from quadratic function.
  • If we input Quadratic Function to quadratic function Quadratic Function , we get an intercept or a crossing point (at Quadratic Function ) between parabola curve and vertical axis. Thus, the third parameter ( Quadratic Function ) determine the intercept point. If Quadratic Function then the parabola will pass the origin Quadratic Function .
  • The extreme point is at coordinate Quadratic Function where Quadratic Function is called discriminant of quadratic equation. We will discuss more about discriminant in the next section . The vertical line passing the extreme point Quadratic Function divides the parabola symmetrically into two parts. Therefore, this vertical line is called axis of symmetry .

Quadratic function may appear in several forms. Below are the forms that are typically found in many applications.

  1. Polynomial form Quadratic Function . This is the general form that we are using throughout the tutorial
  2. Standard form Quadratic Function where Quadratic Function is the extreme point (maximum or minimum point) where Quadratic Function . We will discuss the application example of extreme point in later section . Inputting the coordinate of extreme point to the standard form yield general polynomial form

Quadratic Function

3. Factored form Quadratic Function where Quadratic Function and Quadratic Function are the roots of the quadratic equation where Quadratic Function and Quadratic Function . Inputting the roots into the factored form produces general polynomial form.

In the next section, you will learn about quadratic equation.

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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2008) Quadratic Function Tutorial .
http://people.revoledu.com/kardi/tutorial/quadratic/