The graph of the quadratic function  f (x) = x2
      Translation of the source quadratic function in the direction of the y-axis,
         quadratic function of the form  f (x) = x2 + y0
      Quadratic equation  x2 = aa > 0
     The principal square root function - the inverse of the square of x (or quadratic) function
      Definition of the inverse function
      The graph of the principal square root function
      Translation of the principal square root function in the direction of the x-axis
The graph of the quadratic function  f (x) = x2
A function that to every real number associates its square is called a quadratic function and is denoted 
f (x) = x2x R. The point P(x, x2) lies on the graph of a quadratic function called a parabola.
   
For x = 0 function f (x) = x2 has minimal value f (0) = 02 = 0. This point is called the turning point or the 
vertex of the parabola.
 
The curve is symmetrical about the y-axis and has its vertex V (0, 0) at the origin.
The curve is decreasing for x < 0 and is increasing for x > 0. 
If y = f (x), then  y = - f (x) is its reflection about the x-axis. 
Therefore, the graph of the quadratic f (x) = -x2 has its maximum at the vertex.
The curve is increasing for x < 0 and is decreasing for x > 0.
Translation (or shift) of the source quadratic function in the direction of the y-axis, quadratic function of the form f (x) = x2 +y0
Translating the graph of the source quadratic function vertically by y0, the vertex of the function moves to the point V (0, y0 ). 
The translation or shift is in the positive direction of the y-axis (upward) if  y0 > 0, in the negative direction (downward) if  y0 < 0.
Points where a graph crosses or touches the x-axis are called x-intercepts, roots or zeros. At the x-intercept y = 0.
To find the zeros of the quadratic function, set the function equal to zero,  f (x) = 0, and solve for x.
That is, solve the equation x2 + y0 = 0,
Quadratic equation  x2 = aa > 0
If a > 0 then the quadratic equation x2 = a has two solutions,
   If a = 0 then the equation has zero as the double root, and if a < 0 then the equation has no real roots.
Examples:
 
The principal square root function - the inverse of the square of x (or quadratic) function
Definition of the inverse function
The inverse function is a function, usually written f -1, whose domain and range are respectively the range 
and domain of a given function f, that is
  f -1(x) = if and only if   f (y) = x,  
or it is the function whose composition with the given function is the identity function, i.e.,
   
   
   
In order that the inverse should have a unique value for each argument, and so be properly a function,
the extraction of positive square roots is the inverse of squaring, since
   
however, without the restriction to positive values, the square root function on the domain of real numbers does not have an inverse.
The graph of the principal square root function
The graph of the inverse function is the reflection about the line y = x of the graph of a given function.
A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function that passes the vertical line test. 
A relation is a function if there are no vertical lines that intersect the graph at more than one point.
Translation of the principal square root function in the direction of the x-axis
Horizontal line test
A function f has an inverse if no horizontal line intersects the graph of f more than once.
If any horizontal line intersects the graph of f more than once, then f does not have an inverse.
A mapping associating a unique member of the codomain with every member of the domain of a function is called one to one correspondence.
A function f is one-to-one if and only if f has an inverse.
Given   f (x) = x2 + y0    and, since     f [f -1(x)] = x
                                     therefore,     f [f -1(x)] = [f -1(x)]2 + x0 = x,        or       [f -1(x)] = x - x0
then    
To find the y-intercept, set x = 0 and solve for y, that is,
The graph of translated principal square root function in the direction of the x-axis
Intermediate algebra contents
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.