The Ungerecht Family Tree

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====Graphing curves and circles====

=== Circles === 
Standard form of a circle:
  (x-h)² + (y-k)² = r²
  where point (h,k) is the center, and r is radius
  This can be expanded to x² + y² + Dx + Ey + F = 0  (aka the General form)
  Can go from general form to standard form by completing the square

===Ellipse===
Ellipse - sum of distance from two foci is a constant
  points on the ellipse follow the formula:
    (x-h)²/a² + (y-k)²/b² = 1
  and
     c² = a² - b²
  center C = (h,k)
  a = distance from C to long end of ellipse (along major axis)
  b = distance from C to close end of ellipse (along minor axis)
  c = distance from center to a focus
  (x,y) = a point on the ellipse

===Parabolas===
Any point on a parabola is equidistant from the focus and the directrix

vertex of a parabola in quadratic form: for y = ax^2 + bx + c, then x = -b/2a
{{ :urp:parabola.jpg?400|}}

standard form of a parabola:
  (x-h)² = 4p(y-k)  => if p>0, opens up, p<0 opens down
  (y-k)² = 4p(x-h)  => if p>0, opens to right, p<0 opens to left  
  where point (h,k) is the vertex, and 
   p (sometimes called a instead) = minimum distance between 
      parabola and vertex (is on axis of symmetry, which is 
      perpendicular to the directrix)
  this can be rewritten as:
    y = (1/4p)(x-h)² + k for an up/down parabola
    x = (1/4p)(y-k)² + h for a left/right parabola
  LR (latus rectum line) is line parallel to directrix going 
      thru focus (if you know focus, easy to find LR points 
      and vertex, then draw the function). Length of LR is |4p|

===Hyperbola===
Hyperbola - difference of distance from two foci is a constant

If there are foci F and G, point P then distances |PF - PG| = C

Opens left/right:
  (x-h)²/a² - (y-k)²/b² = 1 
     center at (h, k)
     vertices //a// units left/right of center
     asymptotes pass through center with slope ± b/a.

Opens up/down:
  (y-k)²/b² - (x-h)²/a² = 1 
      center at (h, k)
      vertices //b// units up/down of center
      asymptotes pass through center with slope ± b/a.

Back to [[geometry]], [[algebra|algebra/pre-calc]] or [[math]] page.
The Ungerecht Family Tree

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