Lets study Parabolas by exploring the focus and directrix

There is a chart on page 171 of the Pearson textbook where you can copy formulas and memorize. I plan to show you how to do these problems with just memorizing the vertex form of parabolas. Copy chart below if you wish to see both.

A parabola is the set of all points in a plane that are the same distance from a fixed line and a fixed point not on the line.

The fixed point is called the FOCUS.

The fixed line is called the DIRECTRIX.

Notes: Parabola, Focus, DIrectrix

Note that the distance from the vertex to the focus point is the same as the distance as the vertex to the directrix…

This distance is the “c” value.

To be a parabola……The two red line segments must be the same length!!

Vertex form : with a slight new look since we are studying the focus and directrix

Previous Form using “a” (vertical stretch)

New Form using “c” (focus point)

2axy 2

4

1xc

y

khxay 2 khxc

y 2

4

1

ca

4

1

Opens UP if Opens DOWN if

04

1

0

c

ora

04

1

0

c

ora

Vertex form : How about parabolas that open sideways vs. up and down

Previous Form using “a” (vertical stretch)

New Form using “c” (focus point)

2ayx 2

4

1yc

x

hkyax 2 hkyc

x 2

4

1

ca

4

1

Opens Right if Opens LEFT if

04

1

0

c

ora

04

1

0

c

ora

Write an equation for this parabola!

Vertex is: or

Focus point is at :

Directrix is at:

We could write the equation for this if we knew the value of “c”

Opens up so

kh,

X goes with h and y goes with k

c = distance from vertex to Focus and directrix!!!

2,1 3,1

F

1y

1c

2)1(4

1 2 xy

Now you will graph parabolasExtra hint to help get 2 easy points

on the graph of parabolas

Latus Rectum: move 2c out from the focus to get 2 more points

on the parabola

This will make more sense when we apply it to the next example.

Graph the PARABOLA

vertex :Axis of Symmetry:

382 2 xy

(-3,2)y = 2

Parabola opens: right

Find the focus and directrix :

8

1

4

1c

2c

21X = -5

Latus Rectum = 2c = 4

Vertex Form: 328

1 2 yx

Convert the equation to a parabola in vertex form

025432 yxy

25342 xyy

222 225324 xyy

2132 2 xy

xy 723

1 2

x= a(y-k)2 + h

Find the:

vertex:

focus:

directrix:

(7,2)

3

1

4

1c 4

3c

2,4

37

4

16x

Complete the square

xy 3212 2

Find the vertex, focus and directrix:

242 xxy

Find the:

vertex:

focus:

directrix:

(2,-2)

14

1c

a4

1c

4

31,2

4

12y

Complete the square orTry this:

1 tcoefficienleadinga

22

4

2

a

b2)2(4)2( 2

2

Find the equation of a parabola with a focus (2,-3) and a directrix of x = -4 (the book shows this a different way p.174)

Find the:

direction:

vertex:

c value:

Opens right

)3,1(

3c

),( kh

Sketch to find the direction the parabola opens, the c value, and the vertex. Find a, h, and k to get vertex form

ca

4

1

12

1a 1)3(

12

1 2 yx

HW: Parabolas Due next class!