

The graph of the quadratic function
f
(x)
= x^{2} 
Translation
of the source quadratic function in the direction of the yaxis,
quadratic
function of the form f
(x)
= x^{2 }+ y_{0} 
Quadratic
equation
x^{2}
= a,
a
>
0 





The
graph of the quadratic function
f
(x)
= x^{2} 
A function that to every real number associates its square is called
a quadratic function and is denoted 
f
(x)
= x^{2},
x
Î
R.^{
}The point P(x,
x^{2})
lies on the graph of a quadratic function called a parabola. 

For
x
=
0 function
f
(x)
= x^{2}
has minimal value f
(0)
=
0^{2} =
0.
This point is called the turning point or the
vertex of the parabola.

The
curve is symmetrical about the yaxis 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 xaxis. 
Therefore,
the graph of the quadratic f
(x)
= x^{2
}
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 yaxis, quadratic
function of the form
f
(x)
= x^{2 }+ y_{0} 
Translating
the graph of the source quadratic function vertically by y_{0},
the vertex of the function moves to the point V
(0,
y_{0} ). 
The
translation or shift is in the positive direction of the yaxis
(upward) if y_{0}
> 0, in the negative direction (downward) if y_{0}
< 0. 


Points
where a graph crosses or touches the xaxis
are called xintercepts,
roots or zeros. At the xintercept
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 x^{2 }+ y_{0}
= 0, 



Quadratic
equation
x^{2}
=
a,
a
³
0 
If
a
> 0 then the quadratic equation
x^{2}
=
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. 









College
algebra contents 



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