The constant sum is 2a, the length of the Major Axis.

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The constant sum is 2a, the length of the Major Axis. Distance from Center to Foci is c. Distance from Center to Vertex is a. 2b c + a + a – c = 2a Distance from Center to co-vertex is b. d1 = c + a (0, b) P(x, y) P(x, y) d2 = a – c b d1 c c (-a, 0) a a d2 (a, 0) (-c, 0) (c, 0) 2a b Pythagorean Triangle Relationship. Because the constant sum is 2a, then each red hypotenuse is a units long. (0, -b)

a b b b a a b a

Graph Center @ (0, 0) Find the values of a and b. a2 = 25 a = +5 b2 = 16 b = +4 v Find the values of c to plot the foci. F F V V 3 units away from the center along the Major Axis. v Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +5 +4 Vertices (+5, 0) co-vert. (0, +4) Foci (+3, 0)

Graph Center @ (0, 0) Find the values of a and b. V b2 = 4 b = +2 a2 = 25 a = +5 F Find the values of c to plot the foci. v v F 4.58 units away from the center along the Major Axis. V Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +5 +2 Vertices (0, +5) co-vert. (+2, 0) Foci (0, )

Graph Divide by 36. Center @ (0, 0) Find the values of a and b. V b2 = 4 b = +2 a2 = 9 a = +3 F v v Find the values of c to plot the foci. F V 2.24 units away from the center along the Major Axis. Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +3 +2 Vertices (0, +3) co-vert. (+2, 0) Foci (0, )

Center @ (2, 1) Graph Find the values of a and b. a2 = 36 a = +6 b2 = 9 b = +3 v Find the values of c to plot the foci. F F V V v 5.20 units away from the center along the Major Axis. Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (2, 1) Center @ (2, 1) Center @ (2, 1) +6 +3 Vertices (8, 1), (-4,1) co-vert. (2, 4), (2,-2) Foci ( , 1)

Graph Center @ (-3, 2) V Find the values of a and b. F b2 = 4 b = +2 a2 = 25 a = +5 v v Find the values of c to plot the foci. F V 4.58 units away from the center along the Major Axis. Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (-3, 2) Center @ (-3, 2) Center @ (-3, 2) +5 +2 Vertices (-3, 7),(-3,-3) co-vert. (-1, 2),(-5,2) Foci (-3, )

Graph Complete the square of the x’s and y’s. (+3)2 (-1)2 9 1 v x y x y V V F F Center @ (-3, 1) v Find the values of a and b. a2 = 9 a = +3 b2 = 1 b = +1 Find the values of c to plot the foci. 2.83 units away from the center along the Major Axis. Find the coordinates of the Vertices, co-vertices, and foci. Vertices co-vertices Foci Center @ (-3, 1) Center @ (-3, 1) Center @ (-3, 1) +3 +1 Vertices (0, 1), (-6,1) co-vert. (-3, 2), (-3,0) Foci ( , 1)

Draw a rough graph. Equation format is ... …plug in a & c to solve for b2. c -5 -3 V F 3 5 F V a HORIZONTAL. = 2a Draw a rough graph. 4 = a V 4 V -4 Equation format is ... …plug in a & c to solve for b2. 2 F -2 F c VERTICAL.

Draw a rough graph. Equation format is ... …plug in h, k, a & c to solve for b2. C(2,-2) V(-3,-2) F(4,-2) V(7,-2) a = 7 – 2 = 5 c = 4 – 2 = 2 HORIZONTAL.

Draw a rough graph. Equation format is ... …plug in h, k, a & c to solve for b2. V(2, 5) C(2, 2) V(2, -1) a = 5 – 2 = 3 c = 2 VERTICAL

Draw a rough graph. Equation format is ... …plug in h, k, b & c to solve for a2. v(1, 3) C(1, 2) F(4, 2) c = 4 – 1 = 3 b = 3 – 2 = 1 HORIZONTAL

= c + a – (c – a) d1 = c + a d2 = c – a b d1 c a d2

Graph Center @ (0, 0) Find the values of a and b. a2 = 9 a = +3 b2 = 16 b = +4 v RISE RUN Find the values of c to plot the foci. V V F F 5 units away from the center along the Transverse Axis. v Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +3 +4 Vertices (+3, 0) co-vert. (0, +4) Foci (+5, 0)

Graph Center @ (0, 0) Find the values of a and b. F a2 = 4 a = +2 b2 = 25 b = +5 RISE RUN V v v Find the values of c to plot the foci. V 5.4 units away from the center along the Transverse Axis. F Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +2 +5 Vertices (0, +2) co-vert. (+ 5, 0) Foci (0, )

Graph Set = 1, divide by -36. Center @ (0, 0) F Find the values of a and b. a2 = 9 a = +3 b2 = 4 b = +2 V RISE RUN v v Find the values of c to plot the foci. V F 3.6 units away from the center along the Transverse Axis. Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Center @ (0, 0) Center @ (0, 0) Center @ (0, 0) +3 +2 Vertices (0, +3) co-vert. (+ 2, 0) Foci (0, )

Graph Center @ (2, 1) Find the values of a and b. v a2 = 16 a = +4 b2 = 9 b = +3 RISE F RUN V V F Find the values of c to plot the foci. v 5 units away from the center along the Transverse Axis. Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Point-Slope form Center @ (2, 1) Center @ (2, 1) Center @ (2, 1) +4 +3 Vertices (6, 1),(-2,1) co-vert. (2, 4),(2,-2) Foci (7, 1),(-3,1)

Graph Center @ (-3, 2) Find the values of a and b. a2 = 4 a = +2 b2 = 1 b = +1 v RISE F V F V RUN v Find the values of c to plot the foci. 2.2 units away from the center along the Transverse Axis. Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Point-Slope form Center @ (-3, 2) Center @ (-3, 2) Center @ (-3, 2) +2 +1 Vertices (-1, 2),(-5,2) co-vert. (-3, 3),(-3,1) Foci ( , 2)

4 1 Graph Complete the square. (-2)2 (1)2 F V v v V GCF of -4! 4 1 (-2)2 (1)2 F V v v V Find the values of a and b. Center @ (-1, 2) a2 = 4 a = +2 b2 = 1 b = +1 F RISE RUN Find the values of c to plot the foci. 2.2 units away from the center along the Transverse Axis. Find the coordinates of the Vertices, co-vertices, foci, and asymptote equations. Vertices co-vertices Foci Asymptote Equations Center @ (-1, 2) Center @ (-1, 2) Center @ (-1, 2) +2 +1 co-vert. (0, 2),(-2,2) Vertices (-1, 4),(-1,0) Foci (-1, )

hyperbolas. Draw a rough graph. Equation format is ... …plug in a & c to solve for b2. c a -3 -1 F V 1 3 V F = 2a a = 3 Draw a rough graph. Equation format is ... …plug in a & c to solve for b2. c a 3 5 V F -5 -3 F V HORIZONTAL.

…plug in h, k, a & c to solve for b2. c =2 Draw a rough graph. Equation format is ... …plug in h, k, a & c to solve for b2. c =2 F(3, 7) C(5, 7) V(4, 7) F(7, 7) V(6, 7) a =1 HORIZONTAL. Center = (1, -1) Draw a rough graph. Equation format is ... plug in a, h, k, and use slope to solve for b. V(1, 1) C(1, -1) a = 2 V(1, -3) RISE b = RUN VERTICAL.

… plug in a, c, h, k, and solve for b2. Draw a rough graph. Equation format is ... … plug in a, c, h, k, and solve for b2. F(-3, 6) c = 5 V(-3, 4) a = 3 C(-3, 1) VERTICAL.