M1U3 Conic
Sections
Intro to Conic Sections
Equation
(x – h)2 = 4p(y – k)
(x – h)2 + (y – k)2 = r2
Graph
Name
Parabola
Circle
Ellipse
Hyperbola
Piece of Cone
Parabolas (Locus Definition)
Definition: Set of all points equidistant from a fixed point
(focus) and line (directrix).
focus
directrix
vertex
The midpoint between the focus and directrix is the vertex
Standard Equation of a Parabola (Locus)
Traditional →
Conic
→
y = a ( x - h) + k
( x - h)
2
2
= 4p ( y - k )
In other words, 4p = 1/a!
Here, p is the distance from the focus to the vertex
p=1
Same width as y = x2
→
p = larger →
p = smaller
Wider than y = x2
→
Narrower than y = x2
Or, we can look at the 4p
“4p” is the focal width:
the width of the curve through the focus
Some things to know:
A parabola can go Up/Down or Left/Right
X-squared → Up/Down
Y-squared → Left/Right
You can see the effects of p or 4p on the “steepness”
2
1
2





y

k

a
x

h
 y  k   x  h 
4p
Large p → large 4p → small a → low slope or flatter parabola
Small p → small 4p → large a → high slope or steeper parabola
How can we use this?
• State the vertex, directrix, and focus of the
parabola having the equation (x+3)2 = -20(y-1)
• Vertex:
(-3,1)
• Does the parabola open right or left? Up or down?
4p = -20
• Focus:
• Directrix:
p= -5
Example Find a standard form equation of a parabola with a
vertex at the origin and focus at (0, 8)
Example Find the vertex, focus, and equation of the directrix of
the parabola.
y  x  2x  8 y  9
2
Study Guide Packet
• Due Wednesday
o #1-13 tonight