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Center: (h, k) AOS: x = h and y = k Orientation: Horizontal if a >b Vertical if a<b

Cone cut by a plane at an angle through the sides Real World: Tunnels. Center: (h, k) AOS: x = h and y = k Orientation: Horizontal if a >b Vertical if a<b. Vertices: endpoints of major axis Covertices: endpoints of minor axis Foci: the points c distance from center. Major axis. b. b.

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Center: (h, k) AOS: x = h and y = k Orientation: Horizontal if a >b Vertical if a<b

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  1. Cone cut by a plane at an angle through the sides • Real World: Tunnels • Center: (h, k) • AOS: x = h and y = k • Orientation: Horizontal if a >b Vertical if a<b • Vertices: endpoints of major axis • Covertices: endpoints of minor axis • Foci: the points c distance from center Major axis b b a a a a b b Minor axis Horizontal Ellipse Vertical Ellipse

  2. Steps to graph • Find center • Square root a2 and b2 • Find c • Plot center • Count left and right a spaces • Count up and down b spaces • Connect those 4 with curve. • Count c spaces on major axis both directions to find Foci. Ex 1) Graph Ex 2) Graph • Steps to write equation • Find center • Find a and square it • Find b and square it • Substitute into formula. Center: (-3, 0) a = b= c= Vertices: (-3, 5) (-3, -5) Covertices: (-1, 0) (-5, 0) AOS: x=-3 and y=0 (x-axis) Center: (5, -2) a = b= c= Vertices: (1, -2) (9, -2) Covertices: (5, 1) (5, -5) AOS: x=5 and y=-2 Ex 3) Write the equation given the graph Center: (-1, -2) a = 3 b= 6 Ellipse

  3. Change to Conic Form: Complete the Square • 1 variable squared • 2 variables squared y2 + x + 10y + 26 = 0 y2 + 10y + ___ = -x – 26 + ___ (10/2)=5 then 52 =25 y2 + 10y + 25 = -x – 26 + 25 (y+5)2 = -x – 1 (y+5)2 = -(x +1) x2 + y2 -2x - 4y – 4 = 0 (x2 -2x + ___) + (y2 - 4y + ____) = - 4 + ____ + ____ (-2/2)=-1 then (-1)2=1 AND (-4/2)=-2 then (-2)2=4 (x2 -2x + 1) + (y2 - 4y + 4) = - 4 + 1 + 4 (x-1)2 + (y-2)2 = 1 • GMA (Group, move, add blanks) • Fill in blanks • Factor and simplify • Pre-AP Only • Squared variable coefficient >1 9x2 + 4y2 - 54x - 8y – 59 = 0 9x2 - 54x + 4y2 - 8y = 59 9(x2 - 6x +__) + 4(y2 - 2y + __) = 59 + 9(__) + 4(__) (-6/2)2=9 and (-2/2)2=1 9(x2 - 6x +9) + 4(y2 - 2y + 1) = 59 + 9(9) + 4(1) 9(x – 3)2 + 4(y – 1)2 = 144 6x2 + 12x - y + 15 = 0 6x2 + 12x + __ = y – 15 + ___ 6(x2 + 2x + __) = y – 15 + 6(__) (2/2)=1 then 12=1 6(x2 + 2x + 1) = y – 15 + 6(1) 6(x+1)2 = y – 9 • GMA (Group, move, add blanks) • Factor out GCF and add to blank on other side • Fill in blanks • Factor and simplify

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