1. Circles Ellipse Parabolas Hyperbolas
Conic Sections
Circles
Ellipse
Parabolas
Hyperbolas

2. Trick to looking at an equation and knowing what the shape will look like:
Ellipse – you have 2 squared terms that are added together
Circle – you have 2 squared terms with the same coefficient and are added together
Hyperbola – 2 squared terms that are subtracted
Parabola – 1 square term

3. parabolas

4. PARABOLA FORMULA Opens Opens Vertex (h , k) Vertex (h , k)
Up if p > 0
Down if p < 0
Vertex (h , k)
Directrix: y = k - p
Axis of Symmetry (AOS): x = h
Focus (h , k + p)
Opens
Right if p > 0
Left if p < 0
Vertex (h , k)
Directrix: x = h - p
Axis of Symmetry (AOS): y = k
Focus (h + p , k)

5. To find p you take 4p=4 and solve.
Example 1
Find the required parts for the parabola
𝑦 2 =4𝑥
Rewrite it with all parts in order
𝑦−0 2 =4(𝑥−0)
To find p you take 4p=4 and solve.
p=1
Opens:
right
Vertex:
0,0
Directrix:
x= -1
Axis of Symmetry:
y=0
Focus:
1,0

6. To find p you take 4p=8 and solve.
Example 2
Find the required parts for the parabola
8 𝑦−1 = 𝑥+2 2
Rewrite it with all parts in order
𝑥+2 2 =8(𝑦−1)
To find p you take 4p=8 and solve.
p=2
Opens: up
Vertex:
−2,1
Directrix:
y=-1
Axis of Symmetry:
x=-2
Focus:
−2,3

7. Example 3
Find the required parts for the parabola 𝑥=−3 𝑦 2 +12𝑦−9 𝑥+9=−3 𝑦 2 +12𝑦 𝑥+9=−3( 𝑦 2 −4𝑦+4) 𝑥+9−12=−3 𝑦−2 2 (𝑥−3)=−3 𝑦−2 2 − 1 3 (𝑥−3)= 𝑦−2 2
Your 4p must be on the side without the squared term.
To find p you take 4𝑝=− 1 3 and solve.
𝑝=− 1 12
Opens:
Left
Vertex:
3,2
Directrix:
𝑥= 37 12
Axis of Symmetry:
y=2
Focus:
35 12 ,2

8. Practice Problems x = y² - 6y + 8 y = 4x² y = -x² + 4x – 2 y² + x = 0