Transcript d parabola P F

A parabola is defined as the collection of all
points P in the plane that are the same
distance from a fixed point F as they are
from a fixed line D. The point F is called the
focus of the parabola, and the line D is its
directrix. As a result, a parabola is the set of
points P for which
d(F, P) = d(P, D)
Equation Vertex Focus
y 2  4ax (0, 0) (a, 0)
D: x = -a
Directrix
x = -a
y
V
F = (a, 0)
x
Equation Vertex Focus
y 2  4ax (0, 0) (-a, 0)
y
D: x = a
V
x
F: (-a, 0)
Directrix
x=a
Equation Vertex
x 2  4ay
(0, 0)
Focus
(0, a)
Directrix
y = -a
y
F: (0, a)
V
x
D: y = -a
Equation Vertex Focus
x 2  4ay (0, 0) (0, -a)
Directrix
y=a
y
D: y = a
x
F: (0, -a)
Find an equation of the parabola with vertex
at the origin and focus (-2, 0). Graph the
equation by hand and using a graphing utility.
Vertex: (0, 0); Focus: (-2, 0) = (-a, 0)
y  4ax
2
y  4(2) x
2
y  8 x
2
The line segment joining the two points
above and below the focus is called the
latus rectum.
Let x = -2 (the x-coordinate of the focus)
2
y  8 x
2
y  8( 2)
y  16
2
y  4
The points defining the latus rectum are (-2, -4)
and (-2, 4).
10
(-2, 4)
(0, 0)
10
(-2, -4)
10
0
10
Parabola with Axis of Symmetry Parallel to xAxis, Opens to the Right, a > 0.
Equation
2
 y  k   4a  x  h
Vertex
(h, k)
Focus
(h + a, k)
Directrix
x = -a + h
y D: x = -a + h
Axis of
symmetry
y=k
V = (h, k)
F = (h + a, k)
x
Parabola with Axis of Symmetry Parallel to xAxis, Opens to the Left, a > 0.
Equation
2
 y  k   4 a  x  h 
Vertex
(h, k)
Focus
(h - a, k)
D: x = a + h
y
Axis of
symmetry
y=k
Directrix
x=a+h
F = (h - a, k)
x
V = (h, k)
Parabola with Axis of Symmetry Parallel to yAxis, Opens up, a > 0.
Equation
2
 x  h  4a  y  k 
y
Vertex
(h, k)
Focus
(h, k + a)
Directrix
y = -a + k
Axis of symmetry
x=h
F = (h, k + a)
V = (h, k)
D: y = - a + k
x
Parabola with Axis of Symmetry Parallel to yAxis, Opens down, a > 0.
Equation
2
 x  h   4 a  y  k 
y
Vertex
(h, k)
Focus
(h, k - a)
Directrix
y=a+k
Axis of symmetry
x=h
V = (h, k)
D: y = a + k
F = (h, k - a)
x
Find the vertex, focus and directrix of
2
x + 4x  8y  20  0. Graph the parabola by hand
and using a graphing utility.
x + 4 x  8 y  20  0
2
x + 4 x  8 y + 20
2
x + 4 x + _  8 y + 20
2





2
 4  4
 
 2
x + 4 x + 4  8 y + 20 + 4
2
 x + 2
2
 8 y + 3
 x + 2  8 y + 3
2
 x  h  4a  y  k 
2
Vertex: (h, k) = (-2, -3)
a=2
Focus: (-2, -3 + 2) = (-2, -1)
Directrix: y = -a + k = -2 + -3 = -5
 x + 2  8 y + 3
2
Latus Rectum: Let y = -1
 x + 2  8  1 + 3
2
 x + 2  16
2
x + 2  4
x  6 or x  2
  6,1 or 2,1
10
(-6, -1)
0 (2, -1)
10
y = -5
(-2, -3)
10
(-2, -1)
10