Section 11.1 The Parabola
Objective 1 Determining the Equation of a Parabola with Vertical Axis of Symmetry
We will now look at parabolas from a geometric perspective. When a plane is parallel to an element of a cone,
the plane will intersect the cone in the shape of a parabola.
The set of points that define the parabola formed by the intersection described above is stated in the following
geometric definition of the parabola.
The Geometric Definition of the Parabola
A parabola is the set of all points in a plane equidistant from a fixed point F and a fixed line D.
The fixed point is called the focus and the fixed line is called the directrix.
Axis of Symmetry
Focus
F
P ( x, y )
Vertex
The distance from any point P on the parabola to the focus
is the same as the distance from point P to the directrix.
V
Directrix
We can see that for any point P ( x, y ) that lies on the graph of the parabola, the distance from point P to the
focus is exactly the same as the distance from point P to the directrix. Similarly, because the vertex, V, lies on
the graph of the parabola, the distance from V to the focus must also be the same as the distance from V to the
directrix. Therefore, if the distance from V to F is p units then the distance from V to the directrix is also p
units. If the coordinates of the vertex are ( h, k ) then the coordinates of the focus must be ( h, k  p ) and the
equation of the directrix is y  k  p . We can use this information and the fact that the distance from P ( x, y )
to the focus is equal to the distance from P ( x, y ) to the directrix to derive the equation of a parabola.
The Equation of a Parabola in Standard Form with Vertical Axis of Symmetry
The equation of a parabola with vertical axis of symmetry is ( x  h) 2  4 p( y  k ) where:
The vertex is V (h, k ) .
p  distance from the vertex to focus  distance from the vertex to directirx.
The focus is F (h, k  p ) .
The equation of the directirx is y  k  p .
The parabola opens upward if p  0 or downward if p  0 .
F ( h, k  p )
k
ykp
V ( h, k )
k
V ( h, k )
ykp
F ( h, k  p )
h
h
p0
p0
Objective 2 Determining the Equation of a Parabola with Horizontal Axis of Symmetry
The graph of a parabola could also have a horizontal axis of symmetry and open “sideways”. We derive the
standard form of the parabola with a horizontal axis of symmetry in much the same way as we did with the
parabola with a vertical axis of symmetry.
The Equation of a Parabola in Standard Form with Horizontal Axis of Symmetry
The equation of a parabola with vertical axis of symmetry is ( y  k ) 2  4 p( x  h) where:
The vertex is V (h, k ) .
p  distance from the vertex to focus  distance from the vertex to directrix.
The focus is F (h  p, k ) .
The equation of the directrix is x  h  p .
The parabola opens right if p  0 or left if p  0 .
x  h p
x  h p
V ( h, k )
V ( h, k )
k
k
F ( h  p, k )
F ( h  p, k )
h
p0
h
p0