Today in Precalculus
• Notes: Conic Sections - Parabolas
• Homework
• Test grades will be in home access
before I leave today. We’ll go over
them on Monday.
Conic Sections
Conic sections are formed by the intersection of a plane and a
cone.
circle
ellipse
parabola hyperbola
Degenerate Conic Sections
Atypical conics
point
line
intersecting lines
The conic sections can be defined algebraically in the
Cartesian plane as the graphs of second-degree equations in
two variables, that is, equations of the form: Ax2 + Bxy +
Cy2 + Dx + Ey + F = 0, where A, B, and C are not all zero
Parabolas
Definition: A parabola is the set of all points in a plane
equidistant from the directrix and the focus in the plane.
axis
focus
vertex
directrix
The line passing through the focus and perpendicular to the
directrix is the axis of the parabola and is the line of symmetry
for the parabola.
The vertex is midway between the focus and the directrix and
is the point of the parabola closest to both.
P(x,y)
F(0,p)
By definition, the distance between F
and P has to equal the distance
between P and D.
( x  0) 2  ( y  p) 2  distance between F &P
D(x,-p)
( x  x) 2  ( y  ( p)) 2  distance between P & D
( x  0) 2  ( y  p) 2 
( x - x) 2  ( y  ( p)) 2
x2 +y2 – 2py + p2 = y2 + 2py +p2
x2 = 4py
The standard form of the equation of a parabola that
opens upward or downward
If p>0, the parabola opens upward, if p<0 it opens downward.
Parabolas that open to the left or right are inverse relations of
upward or downward opening parabolas.
So equations of parabolas with vertex (0,0) that open to the
right or to the left have the standard form y2 = 4px
If p>0, the parabola opens to the right and if p<0, the
parabola opens to the left.
p is the focal length of the parabola – the directed distance
from the vertex to the focus of the parabola.
A line segment with endpoints on a parabola is a chord of the
parabola.
The value |4p| is the focal width of the parabola – the length
of the chord through the focus and perpendicular to the axis.
Parabolas with vertex (0,0)
Standard Equation
x2=4py
y2 = 4px
Opens
Up if p>0
Down if p<0
To right if p>0
To left if p<0
Focus
(0, p)
(p, 0)
Directrix
y = -p
x = -p
Axis
y-axis
x-axis
Focal length
p
p
Focal width
|4p|
|4p|
Example 1a
Find the focus, the directrix, and focal width of the
parabola
1 2
y x
12
Example 1b
Find the focus, the directrix, and focal width of the
parabola
x = 2y2
Example 2
Find the equation in standard form for a parabola whose
a) directrix is the line x = 5 and focus is the point (-5, 0)
b) directrix is the line y =6 and vertex is (0,0)
Example 3
Find the equation in standard form for a parabola whose
a) vertex is (0,0) and focus is (0, -4)
b) vertex is (0,0), opens to the left with focal width7
Homework
Page 641: 1,2, 5-20