Quadratic Functions

    Quadratic functions are polynomial functions of degree 2 and are usually studied after linear functions. The method of completing the square is detailed and it is shown that completing the square of the general quadratic equation leads to the quadratic formula. Graphing quadratic equations, including intervals of increasing and decreasing, vertex, line of symmetry, and maximium and minimium values are also illustrated.   

Definition (Quadratic Equations) An equation of the form is called a quadratic equation when are real numbers with

    Some quadratic equations can be solved by merely taking the square root of both sides of the equation, for example can be solved when by taking the square root of both sides of the equation, namely Then solving for we have and so

Example (Quadratic Equations) Solve the quadratic equations:

(a) Solve

    By taking the square root of both sides of the equation we have

(b) Solve

    By taking the square root of both sides of the equation we have and so
    
(c) Solve

    By taking the square root of both sides of the equation we have and so Therefore, and thus and
    
(d) Solve

    Clearly, this equation has no real solution since and

Example (Completing the Square) Solve the following quadratic equations by completing the square.

(a) Solve

  (add -2 to both sides)

(take half of , square, then add)
   
(factor)
  
(perfect square)
  
(square root)
  
  
  
     
  
       

    
    (b) Solve




   

  

  

  
  
  
     


(c) Solve



  
  

   

  

  

  
     
       

Proposition (Quadratic Formula) The quadratic equation has solutions given by

    Proof. By completing the square we have




  

    

  

  
  
  
  
  
  
  
  
  
         

Example (Quadratic Formula) Solve the following equations using the quadratic formula.

(a) Solve

    We have and and the quadratic formula yields,











(b) Solve

    We have and and the quadratic formula yields,








(c) Solve

    We can rewrite this equation so that it is a quadratic equation in namely We have and and the quadratic formula yields,

Definition (Quadratic Functions) Functions of the form where and are real numbers with are called quadratic functions.

     Quadratic functions can be written in standard form Then is a horizontal shift and is a vertical shift. Further, if then the graph is facing up and if then the graph is reflected down.  

Example (Quadratic Functions) Analyze the graphs of the following quadratic functions

(a) Graph



The function has a minimum value of at The function is decreasing on and is increasing on The vertex is the point and the axis of symmetry is


(b) Graph



The function has a maximum value of at The function is increasing on and is decreasing on The vertex is the point and the axis of symmetry is

Example (Completing the Square for Quadratic Functions) Complete the square for each of the functions to rewrite the function in standard form.

(a)

    We have














(b)

    We have

Example (Finding Roots of Quadratic Functions) Finding roots of a quadratic function (or finding the zeros) and solving quadratic equations is the same. For example, solving the equation is the same as finding the roots of the function Either way we can use the quadratic formula.

Find the roots of

    We have and and the quadratic formula yields,

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Cite this as:
Quadratic Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/quadratic-functions.html