x

y

Focus (0,p)

Vertex

(h,k)

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.

P

x

y

Focus (0,p)

Directrix

Vertex

(h,k)

p

p

2p

And the equation

is…

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!

y = -p

42 pyx

x

y

Focus (0,-p)

Directrix y a

Vertex

(h,k)

p

p

And the equation

is…

42 pyx

x

y

Focus (p,0)

Vertex

(h,k)

x aDirectrix p p

2p

And the equation

is…

42 pxy

x

y

Focus (-p,0)

Vertex

(h,k)

x aDirectrix p p

And the equation

is…

42 pxy

STANDARD FORMS

Vertex at ( , )

Opens up

h k

Vertex at ( , )

Opens down

h k

Vertex at ( , )

Opens right

h k

Vertex at ( , )

Opens left

h k

I like to call standard form “Good Graphing Form”

)(4)( )1 2 kyphx

)(4)( )2 2 kyphx

)(4)( )3 2 hxpky

)(4)( )4 2 hxpky

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaStandard Equation of a Parabola (Vertex at Origin)

pyx 42

p ,0focus

yx 122

3 ,0py

directrix3y

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaStandard Equation of a Parabola (Vertex at Origin)

pxy 42

0 ,pfocus

xy 122

0 ,3px

directrix

3x

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

yx 2 .1 2

focus:

2

1 ,0

directrix:2

1y

24 p2

1p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

xy 16 .2 2

focus: 0 ,4

directrix: 4x

164 p 4p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

yx4

1 .3 2

focus:

16

1 ,0

directrix:16

1y

4

14 p

16

1p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

xy 4 .4 2

focus: 0 ,1

directrix: 1x

44 p 1p

1 ,0 .5

Writing an Equation of a ParabolaWriting an Equation of a ParabolaWrite the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

yx 42 pyx 42

1 yx 42

0 ,

2

1 .6

xy 42

pxy 42

2

1 xy 22

4

1 ,0 .8

Writing an Equation of a ParabolaWriting an Equation of a ParabolaWrite the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

yx 42

pyx 42

4

1yx 2

0 ,2 .7

xy 42

pxy 42 2 xy 82

Modeling a Parabolic ReflectorModeling a Parabolic Reflector9. 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.

1 ,5.1

ypx 42 25.1

4

25.2p

p425.2

400

22516

9

1

Notes Over 10.2Modeling a Parabolic ReflectorModeling a Parabolic Reflector10. 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.

yx 42 pyx 42

50yx 2002

2

250

y2001252

125y200625,15

ft 1.78y

General Form of any Parabola

2 2 0Ax By Cx Dy E

*Where either A or B is zero!

* You will use the “Completing the Square” method to go from the

General Form to Standard Form,

Graphing a Parabola: Use completing the square to convert a general form equation to standard

conic form

y2 - 10x + 6y - 11 = 0

9 9y2 + 6y + = 10x + 11 + _____(y + 3)2 = 10x + 20

(y + 3)2 = 10(x + 2)

General form

Standard form

aka: Graphing form

(y-k)2 = 4p(x-h)