2.1 Quadratic Functions

Polynomial Function Definition:

To be a Polynomial Function…

• "n" has to be a non-negative integer. "n" can be a ...

• Whole Number
• 
  
• Positive Integer

• The coefficients have to be real numbers

• The highest value of “n” is the degree where

 Example of a Polynomial Function

• “n” is a Whole Number
• All coefficients are real numbers
• 

The polynomial function described above would be a linear function due to its degree of “1”. If that degree of the function became “2” the polynomial function would become a quadratic function.

Quadratic Function Definition:

Rules

• a, b, and c are real numbers
• 

Example of a Quadratic Function

This quadratic function contains a quadratic term (10x²), linear term (25x), and a constant term(-15).

Standard form of a Quadratic Function

      Standard form of a quadratic function can be found by completing the square of the quadratic function. Having the quadratic function in standard form is very helpful when it comes to graphing parabolas. In standard form it is easy to identify the vertex of a quadratic function.

Completing the Square

Begin with the quadratic: 

Simplify by factoring out a 4:





To complete the square, divide term “b” and multiply the quotient by itself:





Since you can’t just add 4 you must add and subtract 4 to avoid changing the expression:





Now express the equation as a perfect square:





Simplify and distribute the “4” to get Standard Form




From the standard form of this quadratic function you can see that the


x-intercept is (x + 2) being “-2” and the y-intercept is “0”. 


Therefore one is able to find out that the vertex of the quadratic function


  is  from the standard form.