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 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 ³ 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:
 
College algebra contents
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