Wolfram Alpha:

```Parabolas, Ellipses and Hyperbolas
------------------------------------

Standard form:  y = ax2 + bx + c

Vertex form:  y = a(x - h)2 + k

Vertex: (h,k)

Axis of symmetry: x or y = -b/2a

Focus: (h, k + 1/4a) = (r,s)

Equation from directrix and focus:

(x - h)2 = 4p(y - k)

y = (x - r)2/(2(s - yD) + (s + yD)/2

yD = k - 1/4a

Example:

y = 0.05x2 - x + 1

y = 0.05(x2 - 20x + 20)

y = 0.05{(x - 10)2 - 100 + 20}

y = 0.05{(x - 10)2 - 80}

y = 0.05(x - 10)2 - 4

Type:  vertical upward

Axis of symmetry x = -b/2a

x = 1/2*0.05 = 10

y = 0.05*100 - 10 + 1 = 5 - 10 + 1 = -4

Vertex: (10,-4)

Focus: (10,-4 + 1/0.2) = (10,1)

Directrix: y = (-4 - 1/0.2) = -9 ∴ p = 5

Equation from directrix and focus:

(x - h)2 = 4p(y - k)

(x - 10)2 = 20(y + 4)

∴ x2 - 20x + 100 = 20y + 80

∴ x2 - 20x + 20 = 20y

∴ y = 0.05x2 - x + 1

y = (x - r)2/(2(s - yD) + (s + yD)/2

= (x - 10)2/(2(1 - (-9)) + (1 - 9)/2

= (1/20)(x2 - 20x + 100) - 4

= 0.05x2 - x + 1

Ellipse
-------

Center: (h,k)

Horizontal (a > b; as shown):  (x - h)2/a2 + (y - k)2/b2 = 1

Vertical (b > a):  (y - k)2/a2 + (x - h)2/b2 = 1

a, b, c relationship:  c2 = a2 - b2

Example.

(x + 3)2/16 + (y - 2)2/9 = 1

=> Type:  horizontal

Center: (-3,2)

a: 4

b: 3

Vertices: (-3+4,2) = (1,2)
(-3-4,2) = (-7,2)

c: √7

Focii: (-3+√7,2)
(-3-√7,2)

Hyperbola
---------

Center:  (h,k)

Horizontal (as shown):  (x - h)2/a2 - (y - k)2/b2 = 1

∴ y = +/-√(b2(x - h)2/a2 - b2) + k

When x -> ∞ this gives:

y = +/-bx/a + k

When y = 0 this gives:

x = +/-a

Vertical: (y - k)2/a2 - (x - h)2/b2 = 1

y = +/-a(x - h)/b + k

a, b, c relationship:  c2 = a2 + b2

Ex.

(x + 3)2/16 - (y - 2)2/9 = 1

=> Type:  horizontal

Center: (-3,2)

a: 4

b: 3

Vertices: (-3+4,2) = (1,2)
(-3-4,2) = (-7,2)

c: 5

Focii: (-3+5,2) = (2,2)
(-3-5,2) = (-8,2)

Asymptotes: y = +/-3/4(x + 3) + 2

```