Notes 8.1
Conics Sections –
The Parabola
I. Introduction
A.) A conic section is the intersection of
a plane and a cone.
B.) By changing the angle and the
location of intersection, a parabola,
ellipse, hyperbola, circle, point, line, or
a pair of intersecting lines is
produced.
C.) Standard Conics:
1.) Parabola
2.) Ellipse
3.) Hyperbola
D.) Degenerate Conics
1.) Circle
2.) Point
3.) Line
4.) Intersecting Lines
E.) Forming a Parabola –
When a plane intersects a double-napped
cone and is parallel to the side of the cone,
a parabola is formed.
F.) General Form Equation for All Conics
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
If both B and C = 0, or A and B = 0, the
conic is a parabola
II. The Parabola
A.) In general - A parabola is the graph of
a quadratic equation, or any equation in
the form of
y  ax  bx  c
2
or
x  ay  by  c
2
B.) Def. - A PARABOLA is the set of
all points in a plane equidistant from a
particular line (the DIRECTRIX) and a
particular point (the FOCUS) in the
plane.
Axis of Symmetry
Focus
Focal Width
Vertex
Focal Length
Directrix
C.) Parabolas (Vertex = (0,0))
Standard Form
Focus
Directrix
x  4 py
y  4 px
 0, p 
 p,0
2
y  p
Axis of Symmetry y  axis
Focal Length
Focal Width
2
x  p
x  axis
p
p
4p
4p
D.) Ex. 1- Find the focus, directrix, and focal
width of the parabola y = 2x2.
1
2
x  y
2
 1
Focus =  0, 
 8
1
 4p
2
1
Directrix = y  
8
1
p
8
Focal Width =
1
2
1 2
E.) Ex. 2- Do the same for the parabola x   y
4
 1, 0 
y  4 x
Focus =
4  4 p
Directrix =
p  1
Focal Width =
2
x 1
4
F.) Ex. 3- Find the equation of a parabola with a
directrix of x = -3 and a focus of (3, 0).
4 px  y
2
4  3 x  y
12x  y
2
2
G.) Parabolas (Vertex = (h, k))
St. Fm.
 x  h
2
 4p y  k
y k
2
 4 p  x  h
Focus
 h, k  p 
 h  p, k 
Directrix
ykp
x  h p
Ax. of Sym.
xh
yk
Fo. Lgth.
p
Fo. Wth.
4p
p
4p
H.) Ex. 4- Find the standard form equation for
the parabola with a vertex of (4, 7) and a focus
of (4, 3).
ax. of sym. 
 x4
 x  h
2
 x  4
 4p y  k
2
 x  4
 4 p  y  7
2
 16  y  7 
p  3  7  4
I.) Ex. 5- Find the vertex, focus, and directrix of
the parabola 0 = x2 – 2x – 3y – 5.
x  2x  3 y  5
2
5

focus =  1,  
4

x  2x  1  3 y  5  1
2
 x  1
2
vertex =
 3 y  2
1, 2
Directrix =
11
y
4
III. Paraboloids of Revolution
A.) A PARABOLOID is a 3-dimensional
solids created by revolving a parabola
about an axis.
Examples of these include headlights,
flashlights, microphones, and satellites.
B.) Ex. 6– A searchlight is in the shape of a
paraboloid of revolution. If the light is 2
feet across and 1 ½ feet deep, where
should the bulb be placed to maximize
the amount of light emitted?
x  4 py
2
3
1  4 p  
2
3

 1, 
2

 0, p 
2
 0, 0 
1
p
6
The bulb should be placed 2”
from the vertex of the paraboloid
 3
1, 
 2