1. 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)

2. a b b b a a b a

3. Graph
(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
(0, 0)
(0, 0)
(0, 0) +5 +4
Vertices (+5, 0)
co-vert. (0, +4)
Foci (+3, 0)

4. Graph
(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
(0, 0)
(0, 0)
(0, 0) +5 +2
Vertices (0, +5)
co-vert. (+2, 0)
Foci (0, )

5. Graph
Divide by 36.
(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
(0, 0)
(0, 0)
(0, 0) +3 +2
Vertices (0, +3)
co-vert. (+2, 0)
Foci (0, )

6. (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
(2, 1)
(2, 1)
(2, 1) +6 +3
Vertices (8, 1), (-4,1)
co-vert. (2, 4), (2,-2)
Foci ( , 1)

7. Graph
(-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
(-3, 2)
(-3, 2)
(-3, 2) +5 +2
Vertices (-3, 7),(-3,-3)
co-vert. (-1, 2),(-5,2)
Foci (-3, )

8. 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
(-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
(-3, 1)
(-3, 1)
(-3, 1) +3 +1
Vertices (0, 1), (-6,1)
co-vert. (-3, 2), (-3,0)
Foci ( , 1)

9. Draw a rough graph.
Equation format is ...
…plug in a & c to solve for b2. c
V F
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.

10. 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.

11. 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

12. 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

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

15. Graph
(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
(0, 0)
(0, 0)
(0, 0) +3 +4
Vertices (+3, 0)
co-vert. (0, +4)
Foci (+5, 0)

16. Graph
(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
(0, 0)
(0, 0)
(0, 0) +2 +5
Vertices (0, +2)
co-vert. (+ 5, 0)
Foci (0, )

17. Graph
Set = 1, divide by -36.
(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
(0, 0)
(0, 0)
(0, 0) +3 +2
Vertices (0, +3)
co-vert. (+ 2, 0)
Foci (0, )

18. Graph
(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
(2, 1)
(2, 1)
(2, 1) +4 +3
Vertices (6, 1),(-2,1)
co-vert. (2, 4),(2,-2)
Foci (7, 1),(-3,1)

19. Graph
(-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
(-3, 2)
(-3, 2)
(-3, 2) +2 +1
Vertices (-1, 2),(-5,2)
co-vert. (-3, 3),(-3,1)
Foci ( , 2)

20. 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.
(-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
(-1, 2)
(-1, 2)
(-1, 2) +2 +1
co-vert. (0, 2),(-2,2)
Vertices (-1, 4),(-1,0)
Foci (-1, )

21. hyperbolas.
Draw a rough graph.
Equation format is ...
…plug in a & c to solve for b2. c a
F V
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
F V
HORIZONTAL.

22. …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.

23. … 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.