2 2

Write an equation of a circle passing thr

Draw the circle (rough sketch) and give the

ough the given point

and has a center at th

1

e

.

Write the equa

origi

( 2) ( 3) 36

2. ( 8,6)

center and

n

ti

radius.

.

on

x y

2 2

of the circle in standard form.

State the Ce

3.

nter and

14 2 49 0

Radi

us

x y x y

Warm-up!!

CCGPS GeometryDay 60 (11-5-13)

UNIT QUESTION: How are the equations of circles and parabolas derived?Standard: MCC9-12..A.REI.7, G.GPE.1,2 and 4

Today’s Question:How do we graph a parabola from a given equation in standard form?Standard: MCC9-12..G.GPE.2

Parabolas

Parabolas

Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix.

Directrix

The light source is theFocus

The cross section of a headlightis an example of a parabola...

Here are some other applications of the parabola...

Directrix

Focus

d1

d1

d2

d2

d3

d3

Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix...

We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola....

Vertex

Notice that the vertex is located at the midpoint between the focusand the directrix...

Standard Equation of a Parabola: (Vertex at the origin)

Equation Focus Directrix

x2 = 4py (0, p) y = –p

Equation Focus Directrix

y2 = 4px (p, 0) x = –p

(If the x term is squared, the parabola is up or down)

(If the y term is squared, the parabola isleft or right)

Tell whether the parabola opens up down, left, or right.

2

2

2

. 5

2 8

. 4

.

A

B y x

x

C y

y

x

down

right

left

Find the focus and equation of the directrix. Then sketch the graph.

21. 16y x

: ,0Focus p 4 16p

4,0

4p

:Directrix x p

4x

Opens right

Find the focus and equation of the directrix. Then sketch the graph.

22. 2x y : 0,Focus p

4 2p 1

0,2

1

2p

:Directrix y p1

2y

Opens up

Find the focus and equation of the directrix. Then sketch the graph.

23. 12x y

: 0,Focus p4 12p

0, 3

3p

:Directrix y p

3y

Opens down

Find the focus and equation of the directrix. Then sketch the graph.

24. 3 12 0 y x

: 0,Focus p4 4p

1,0

1p

:Directrix y p

1x

Opens left

Example 5: Determine the focus and directrix of the parabola (y – 2)2 = -16 (x - 5) :

Direction:

Vertex:

Focus:

Directrix:

Example 6: Determine the focus and directrix of the parabola (x – 6)2 = 8(y + 3) :

Direction:

Vertex:

Focus:

Directrix:

7. Write the equation in standard form by completing the square. State the VERTEX.

2 2 8 17 0 x x y

8. Write the equation in standard form by completing the square. State the VERTEX.2 6 2 9 0y y x

You TRY!!