2. Geometric Definition of a Parabola: The collection of
all the points P(x,y) in a a plane that are the same
distance from a fixed point, the focus, as they are from
a fixed line called the directrix.
y
Focus
(0,p) P
x
Verte
x
(h,k)

3. y As you can plainly see
the distance from the
focus to the vertex is a
and is the same distance
from the vertex to the
directrix! Neato!
Focus
(0,p) p
2p x
Verte
x p
Directrix y = -p
(h,k)
And the
equation x = 4 py
2
is…

4. y
Directrix = a
y
Verte p
x x
(h,k)
p
Focus
(0,-p)
And the
equation
x = −4 py
2
is…

5. y
2p
Directrix = −a
x p p
Verte Focus
x (p,0)
x
(h,k)
And the y = 4 px
2
equation

6. y
p p
Vertex
Directrix = a
x
Focus (- (h,k)
p,0)
x
And the y = −4 px
2
equation

7. STANDARD FORMS
Vertex at (h, k )
1) ( x − h) 2 = 4 p ( y − k )
Opens up
Vertex at (h, k )
2) ( x − h) 2 = −4 p ( y − k )
Opens down
Vertex at (h, k )
3) ( y − k ) 2 = 4 p ( x − h)
Opens right
Vertex at (h, k )
4) ( y − k ) = −4 p ( x − h)
2
Opens left
I like to call standard form “Good Graphing Form”

8. Graphing an Equation of a Parabola
Standard Equation of a Parabola (Vertex at Origin)
x = 4 py
2
x = 12 y
2
( 0, p )
focus ( 0, 3)
y = −p
directrix
y = −3

9. Graphing an Equation of a Parabola
Standard Equation of a Parabola (Vertex at Origin)
y = 4 px
2
y = 12 x
2
( p, 0 )
focus ( 3, 0 )
x = −p
directrix
x = −3

10. Graphing an Equation of a Parabola
Graph the equation. Identify the focus and directrix of the
parabola.
1. x = 2 y
2
1
4p = 2 p =
2
 1
focus:  0, 
 2
1
directrix: y = −
2

11. Graphing an Equation of a Parabola
Graph the equation. Identify the focus and directrix of the
parabola.
2. y = 16 x
2
4 p = 16 p = 4
focus: ( 4, 0 )
directrix: x = −4

12. Graphing an Equation of a Parabola
Graph the equation. Identify the focus and directrix of the
parabola.
1
3. x = − y
2
1 4 1
4p = − p = −
4 16
 1
focus:  0, − 
 16 
1
directrix: y =
16

13. Graphing an Equation of a Parabola
Graph the equation. Identify the focus and directrix of the
parabola.
4. y = −4 x
2
4 p = −4 p = −1
focus: ( − 1, 0 )
directrix: x = 1

14. Writing an Equation of a Parabola
Write the standard form of the equation of the parabola
with the given focus and vertex at (0, 0).
5. 0, 1( ) x = 4 py
2
x = 4( 1 ) y
2
x = 4y
2
 1 
6.  − , 0  y = 4 px
2
 2 
y 2
=4−
1
2
( )x y = −2 x
2

15. Writing an Equation of a Parabola
Write the standard form of the equation of the parabola
with the given focus and vertex at (0, 0).
7. ( − 2, 0 ) y = 4 px
2
y = 4( − 2) x y = −8 x
2 2
 1
8.  0,  x = 4 py
2
 4
1
x = 4(
2
4
)y x =y 2

16. Modeling a Parabolic Reflector
9. A searchlight reflector is designed so
that a cross section through its axis is a
parabola and the light source is at the
focus. Find the focus if the reflector is 3
feet across at the opening and 1 foot deep.
(1x5) = 4 p (1)
. 2 2
y
2.25 = 4 p
(1.5, 1) p = 2.25 = 9
225
4
400 16

17. Notes Over 10.2
Modeling a Parabolic Reflector
10. One of the largest radio telescopes has
a diameter of 250 feet and a focal length
of 50 feet. If the cross section of the radio
telescope is a parabola, find the depth.
x = 4 py
2 250
= 125
x = 4( 50 ) y
2
2
x = 200 y 15,625 = 200 y
2
125 = 200 y
2
y = 78.1 ft

18. General Form of any Parabola
Ax + By + Cx + Dy + E = 0
2 2
*Where either A or B is zero!
* You will use the “Completing the
Square” method to go from the
General Form to Standard Form,

19. Graphing a Parabola: Use completing the square
to convert a general form equation to standard
conic form
General form
y2 - 10x + 6y - 11 = 0
y2 + 6y + 9 = 10x + 11 + _____
9
(y + 3)2 = 10x + 20
(y + 3)2 = 10(x + 2)
Standard form
aka: Graphing form
(y-k)2 = 4p(x-h)