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 - 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
Any point on a parabola is equidistant from the focus and the directrix
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 - difference of distance from two foci is a constant
If there are foci F and G, point P then distances |PF - PG| = C
(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.
(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.