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 y-axis
         Equations of the parabola written in the general form
      Parabola, examples
Transformation of the equation of a parabola
The equation  y2 = 2pxp < 0  represents the parabola opens to the left since must be y2 > 0. Its axis of symmetry is the x-axis.
If variables x and y change the role obtained is the parabola whose axis of symmetry is y-axis.
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 = ax2, 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  y2 = 2px in the direction of the coordinate axes the vertex of the parabola can be brought at a point A(x0, y0) while coordinates x and y of every point of the parabola changes by the 
value of the translations x0 and y0, so that equation of the translated parabola is
  (y - y0)2 = 2p(x - x0)  
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 - y0)2 = 4a(x - x0).                          
The translated parabola with the axis parallel to the x-axis  can also be written as
x = ay2 + by + c.
The parabola whose axis of symmetry is parallel to the y-axis
By translating the parabola  x2 = 2py  its vertex is moved  from the origin to the point A(x0, y0) so that its equation
transforms to (x - x0)2 = 2p(y - y0).                             
The axis of symmetry of this parabola is parallel to the       y-axis.
As we already mentioned, this parabola is a function that   we usually write
  y = ax2 + bx + c  or  y - y0 = a(x - x0)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 x-axis  b) the axis of the parabola parallel to the y-axis
    Ay2 + Bx + Cy + D = 0A and B not 0,      Ax2 + Bx + Cy + D = 0A and C not 0
      or   x = ay2 + by + ca not 0.         or   y = ax2 + bx + ca not 0.
Note that the parabola has equation that contains only one squared term.
Parabola, examples
Example:  Write equation of the parabola y2 = 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)   =>    y2 = 2px
                         42 = 2p(-4)   =>   p = -2
thus, the equation of the parabola   y2 = -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 = -x2 + 6x - 7.
Solution:   Rewrite the equation of the parabola in the translatable form
    (x - x0)2 = 2p(y - y0or   y - y0 = a(x - x0)2
so,  y = -x2 + 6x - 7  =>   y = -(x2 - 6x) - 7
                                          y = -[(x - 3)2 - 9] - 7
 y - y0 = a(x - x0)2,    y - 2 = -(x - 3)2a = -1.
The vertex of the parabola  A(x0, y0),  or  A(3, 2).
The focus  F(x0, y0 + 1/(4a)),  or  F(3, 7/4).
The equation of the directrix,
y = y0 - 1/(4a),    y = 2 + 1/4  or  y = 9/4.
College algebra contents E
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