Quadratic Equations and Quadratic Function
      Quadratic function or the second-degree polynomial
         Translated form of quadratic function
         Vertex (maximum/minimum) - coordinates of translation
         Roots or zeros of the function, axis of symmetry and y-intercept
      Graphing a quadratic function
         Transformations of the graph of the quadratic function
Quadratic function or the second-degree polynomial
The polynomial function of the second degree,  f (x) = a2x2 + a1x + a0 is called a quadratic function.
   y = f (x = a2x2 + a1x + a0   or   y - y0 = a2(x - x0)2,  
where  are the coordinates of translations of the quadratic 
function. By setting   x0 = 0 and  y0 = we obtain  y = a2x2,  the source quadratic function.
The turning point  V (x0, y0) is called the vertex of the parabola.
Note that the coefficients, a2, a1 and a0, of quadratic function, correspond to the coefficients, a, b and c, of
quadratic equation, respectively.
The real zeros of the quadratic function:  
The above formula is known quadratic formula that shows the symmetry of the roots relative to the axis of 
symmetry of the parabola.
   y = f (x= a2x2 + a1x + a0  = a2(x - x1)(x - x2) = a2[x2 - (x1 + x2)x + x1x2]  
The graph of a quadratic function is curve called a parabola. The parabola is symmetric with respect to a vertical line called the axis of symmetry.
As the axis of symmetry passes through the vertex of the parabola its equation is x = x0.
Quadratic function has the y-intercept at the point ( 0, a0 ).
Translated form of quadratic function
The proof that quadratic function  f (x) = a2x2 + a1x + a0 is translation of its source or original  f (x) = a2x2
1)  Let calculate the coordinates of translations of quadratic function using the formulas,
substitute n = 2 in    
then    
2)  To get the source quadratic function we should plug the coordinates of translations (with changed signs)
     into the general form of the quadratic, i.e.,
after expanding and reducing obtained is
                                     y = a2x2   the source quadratic function
3)  Inversely, by plugging the coordinates of translations into the source quadratic function
                                     y - y0 = a2(x - x0)2,
   
and after expanding and reducing we obtain
                                     y = a2x2 + a1x + a0   the quadratic function in the general form.
Graphing a quadratic function
Transformations of the graph of the quadratic function
How changes in the expression of the quadratic function affect its graph is shown in the figures below.
    The graph of quadratic polynomial will intersect the x-axis in two distinct points if its leading coefficient   a2  and the vertical translation y0 have different signs, i.e., if   a2 y0 < 0. 
Example:  Find zeros and vertex of the quadratic function  y = - x2 + 2x + 3  and sketch its graph.
Solution:  A quadratic function can be rewritten into translatable form  y - y0 = a2(x - x0)2  by completing the square,
      y = - x2 + 2x + 3    Since a2 y0 < 0 given quadratic function must have two different real zeros.
      y = - (x2 - 2x) + 3   To find zeros of a function, we set y equal to zero and solve for x. Thus,
      y = - [(x - 1)2 - 1] + 3                         - 4 = - (x - 1)2
y - 4 = - (x - 1)2                  (x - 1)2 = 4
y - y0 = a2(x - x0)2                      x - 1 = sqrt(4)
V(x0, y0)  =>   V(1, 4)                        x1,2 = 1 2,   =>    x1 = - 1 and  x2 = 3.
We can deal with the given quadratic using the property of the polynomial explored under the title,
' Source or original polynomial function '. Thus,
1)  calculate the coordinates of translations of the quadratic  y = f (x= - x2 + 2x + 3
2)  To get the source quadratic function, plug the coordinates of translations (with changed signs)
     into the general form of the quadratic, i.e.,
y + y0 = a2(x + x0)2 + a1(x + x0) + a0   =>    y + 4 = - (x + 1)2 + 2(x + 1) + 3
                                                                                              y = - x2   the source quadratic function
3)  Inversely, by plugging the coordinates of translations into the source quadratic function
y - y0 = a2(x - x0)2   =>     y - 4 = - (x - 1)2
                    obtained is given quadratic in general form     y = - x2 + 2x + 3.
Intermediate algebra contents
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