
Conic
Sections 


Parabola

Transformation of the equation of a parabola

Equation of a translated parabola  the standard form

The parabola whose axis
of symmetry is parallel to the yaxis 
Equations of the parabola
written in the general form 
Parabola, examples






Transformation of the equation of a parabola

The equation
y^{2} = 2px,
p < 0
represents the parabola opens to the left since must be y^{2}
> 0. Its axis of
symmetry is the xaxis. 
If variables
x and
y change the role obtained is the parabola whose axis of
symmetry is yaxis.

For
p >
0
the parabola opens up, if
p <
0
the parabola opens down as shows
the below figure. 

This parabola we often write
y = ax^{2},
where
a = 1/(2p) , with the focus at
F(0,
1/(4a)) and the directrix 
y =
1/(4a).
This parabola is a function since a vertical line crosses its graph at only one point.


Equation of a translated parabola  the standard form

By parallel shifting of the parabola
y^{2} = 2px in the direction of the coordinate axes the vertex of the parabola
can be brought at a point A(x_{0}, y_{0}) while coordinates
x
and y of every point of the parabola changes by the 
value of the
translations x_{0} and
y_{0}, so that equation of the translated
parabola is


(y

y_{0})^{2} = 2p(x

x_{0}) 


If p >
0
the parabola
opens to the right and if p
< 0
the
parabola opens to the left.

If the focal parameter
p = 2a
then the above equation

becomes,
(y

y_{0})^{2} =
4a(x

x_{0}).

The translated parabola with the axis parallel to the
xaxis
can also be written as

x =
ay^{2}
+ by
+ c.





The parabola whose axis of symmetry is parallel to the yaxis 
By translating the parabola
x^{2} = 2py
its vertex is moved from the origin to the point A(x_{0}, y_{0})
so that its equation

transforms to
(x

x_{0})^{2} = 2p(y

y_{0}).

The axis of symmetry of this parabola is parallel to the
yaxis.

As we already
mentioned, this parabola is a function that we usually
write


y = ax^{2}
+ bx + c
or y

y_{0} =
a(x

x_{0})^{2}, 


where 



This parabola opens up if
a >
0 and opens down if
a < 0.





Equations of the parabola written in the general form 
a) the axis of the parabola parallel to the
xaxis


b) the axis of the parabola parallel to the
yaxis 
Ay^{2}
+ Bx + Cy +
D =
0, A and
B not
0, 

Ax^{2}
+ Bx + Cy +
D =
0, A and
C not 0 
or x =
ay^{2}
+ by
+ c,
a not 0. 

or
y =
ax^{2}
+ bx
+ c,
a not 0. 

Note
that the parabola has equation that contains only one squared term. 

Parabola,
examples 
Example:
Write equation of the parabola
y^{2} = 2px
passing through the point P(4,
4) and find the focus,
the equation of the directrix and draw its graph. 

Solution:
The coordinates of the point P
must satisfy the equation of the parabola

P(4,
4)
=>
y^{2} =
2px

4^{2} = 2p(4)
=> p =
2

thus, the
equation of the parabola y^{2} =
4x.

The coordinate
of the focus,

since F(p/2,
0) then
F(1,
0).

The equation of the directrix,
as x =
 p/2,
x =
1.





Example:
Find the vertex, the focus and the equation of the directrix and draw the graph of the parabola 
y =
x^{2}
+ 6x 
7. 
Solution:
Rewrite
the equation of the parabola in the translatable form

(x

x_{0})^{2} = 2p(y

y_{0}) or
y

y_{0} = a(x

x_{0})^{2}

so, y =
x^{2}
+ 6x 
7
=> y =
(x^{2}

6x) 
7

y =
[(x

3)^{2}

9] 
7

y

y_{0} = a(x

x_{0})^{2},
y

2 = (x

3)^{2},
a
= 1.

The vertex of the parabola
A(x_{0}, y_{0}),
or
A(3, 2).

The focus
F(x_{0},
y_{0 }+ 1/(4a)),
or
F(3,
7/4).

The equation of the directrix,

y =
y_{0 }
1/(4a),
y = 2 +
1/4 or
y =
9/4.













College
algebra contents E 



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