Conic Sections
Parabola Definition and construction of the parabola  Transformation of the equation of a parabola Graphs of the parabola examples
Definition and construction of the parabola
A parabola is the set of all points in a plane that are the same distance from a point F, called the focus, and a line d, called the directrix.
 A parabola is uniquely determined by the distance of the    focus from the directrix. This distance is called the focal parameter p and its          midpoint A is the vertex (apex) of the parabola. The parabola has an axis of symmetry which passes         through the focus perpendicular to the directrix. The distance of any point P of the parabola from the          directrix and from the focus is denoted r, so d(Pd P) = FP = r. Construction of the parabola To a given parameter p of the parabola draw corresponding directrix d and the focus F. To the distances greater then p/2 draw lines, of arbitrary    dense, parallel to the directrix. Then, intersect each line at two symmetric points by arc    centered at the focus with a radius which equals the distance of that line from the directrix. The distance p/2 from the vertex A to the directrix or focus is called focal distance. Vertex form of the equation of a parabola
If a parabola is placed so that its vertex coincides with the origin of the coordinate system and its axis lies
along the
x-axis then for every point of the parabola According to definition of the parabola  FP = d(Pd P) = r after squaring or
 y2 = 2px the vertex form of the equation of the parabola,
 or y2 = 4ax if the distance between the directrix and
focus is given by  p = 2a.
The explicit form of the equation  y = ± Ö2px
shows that to every positive value of x correspond two         opposite values of y which are symmetric relative to the      x-axis. The parabola   y2 = 2px,  p > 0 is not defined for x < 0, it opens to the right. For x = p/2 the corresponding ordinate y = ± p. This parabola is not a function since the vertical line          crosses the graph more than once. A focal chord is a line segment passing through the focus  with endpoints on the curve. The latus rectum is the focal chord (P1P2 = 2p)               perpendicular to the axis of the parabola. Therefore, we can easily sketch the graph of the parabola   using following points, A(0, 0), P1,2(p/2, ± p) and P3,4(2p, ±2p) Graphs of the 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:  Into a parabola y2 = 2px inscribed is an equilateral triangle whose one vertex coincides with the vertex of the parabola and whose area A = 243Ö3. Determine equation of the parabola and remaining vertices of the triangle.
 Solution:   Let write coordinates of a point P of the  parabola as elements of the equilateral triangle As the point P lies on the parabola then The area of the equilateral triangle we express by              coordinates of P  Show the parameter of the parabola by the side of the triangle, and the vertices of the triangle P(6p, 2Ö3p) and P'(6p, -2Ö3pso that,  P(27, 9Ö3) and P'(27, -9Ö3).
Therefore, the equation of the parabola  y2 = 2px  or  y2 = 9x.
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 or general form (x - x0)2 = 2p(y - y0)  or   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)2,  a = -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.    Intermediate algebra contents 