

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 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
xaxis 





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
>
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. 


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)
= y 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 xaxis 
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 onetoone if and only if f
has an inverse. 
Given
f
(x)
= x^{2} + y_{0}_{
}and,
since f
[f
^{}^{1}(x)]
= x 
therefore, f
[f
^{}^{1}(x)]
= [f
^{}^{1}(x)]^{2
}+ x_{0}
= x, or
[f
^{}^{1}(x)]^{2 }
= x 
x_{0}

then 



To find the
yintercept,
set x
= 0 and solve for y, that is, 



The
graph of
translated principal square root function
in the direction of
the xaxis 









Intermediate
algebra contents 



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