Warm-up!!
Draw the circle (rough sketch) and give the center and radius.
1. ( x  2) 2  ( y  3) 2  36
Write an equation of a circle passing through the given point
and has a center at the origin.
2. (8, 6)
Write the equation of the circle in standard form.
State the Center and Radius
3. x 2  y 2  14 x  2 y  49  0
CCGPS Geometry
Day 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.
The cross section of a headlight
is an example of a parabola...
Directrix
The light
source is the
Focus
Here are some other applications of the
parabola...
d2
d1Focus
d3
d1
Vertex d3
d2
Directrix
Notice that the vertex is located at the midpoint between the focus
and the directrix...
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....
Standard Equation of a Parabola: (Vertex at the origin)
Equation
2
x = 4py
Focus
Directrix
(0, p)
y = –p
(If the x term is squared, the parabola
is up or down)
Equation
2
y = 4px
Focus
Directrix
(p, 0)
x = –p
(If the y term is squared, the parabola is
left or right)
Tell whether the parabola opens
up down, left, or right.
A.
2
y 5x
B.
2 y 2  8x
right
C.
4x  y2
left
down
Find the focus and equation of the
directrix. Then sketch the graph.
1. y  16 x
2
 4, 0 
Focus :  p,0  4 p  16
p4
Directrix : x   p
x  4
Opens right
Find the focus and equation of the
directrix. Then sketch the graph.
2. x  2 y
2
Focus :  0, p 
4p  2
1
p
2
Directrix : y   p
Opens up
 1
 0, 
 2
1
y
2
Find the focus and equation of the
directrix. Then sketch the graph.
3. x  12 y
2
Focus :  0, p 
 0, 3
4 p  12
p  3
Directrix : y   p
y 3
Opens down
Find the focus and equation of the
directrix. Then sketch the graph.
4.  3 y  12 x  0
2
Focus :  0, p 
4 p  4
p  1
Directrix : y   p
x  1  1,0
Opens left
Example 5: Determine the focus and directrix of the
2
parabola (y – 2) = -16 (x - 5) :
Direction:
Vertex:
Focus:
Directrix:
Example 6: Determine the focus and directrix of the
2
parabola (x – 6) = 8(y + 3) :
Direction:
Vertex:
Focus:
Directrix:
7. Write the equation in standard form by
completing the square. State the VERTEX.
2
x  2 x  8y  17  0
You TRY!!
8. Write the equation in standard form by
completing the square. State the VERTEX.
2
y  6y  2x  9  0