1. Graph an equation of a hyperbola
EXAMPLE 1
Graph an equation of a hyperbola
Graph 25y2 – 4x2 = Identify the vertices, foci, and asymptotes of the hyperbola.
SOLUTION
STEP 1
Rewrite the equation in standard form.
25y2 – 4x2 = 100
Write original equation.
25y2
100 –
4x2 =
Divide each side by 100. y2 4 –
y225
= 1
Simplify.

2. EXAMPLE 1
Graph an equation of a hyperbola
STEP 2
Identify the vertices, foci, and asymptotes. Note that a2 = 4 and b2 = 25, so a = 2 and b = 5. The y2 - term is positive, so the transverse axis is vertical and the vertices are at (0, +2). Find the foci.
c2 = a2 – b2 = 22 – = 29.
so c =
29.
The foci are at ( 0, )
29.
(0, + 5.4).
The asymptotes are y = ab + x or 25
y =

3. EXAMPLE 1
Graph an equation of a hyperbola
STEP 3
Draw the hyperbola. First draw a rectangle centered at the origin that is 2a = 4 units high and 2b = 10 units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.

4. EXAMPLE 2
Write an equation of a hyperbola
Write an equation of the hyperbola with foci at (–4, 0) and (4, 0) and vertices at (–3, 0) and (3, 0).
SOLUTION
The foci and vertices lie on the x-axis equidistant from the origin, so the transverse axis is horizontal and the center is the origin. The foci are each 4 units from the center, so c = 4. The vertices are each 3 units from the center, so a = 3.

5. Write an equation of a hyperbola
EXAMPLE 2
Write an equation of a hyperbola
Because c2 = a2 + b2, you have b2 = c2 – a2. Find b2.
b2 = c2 – a2 = 42 – 32 = 7
Because the transverse axis is horizontal, the standard form of the equation is as follows: x2 32 –
y2 7
= 1
Substitute 3 for a and 7 for b2. x2 9 –
y2 7
= 1
Simplify

6. GUIDED PRACTICE
for Examples 1 and 2
Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola. x2 16 –
y2 49 1.
= 1
SOLUTION 74 + x
y =
, ( ) ,
65 , 0
(+4, 0)

7. GUIDED PRACTICE
for Examples 1 and 2 y2 36 2.
– x2
= 1
SOLUTION
( 0, ) , 37
(0, +6) ,
y = +6x

8. GUIDED PRACTICE
for Examples 1 and 2
y2 – 9x2 = 36
SOLUTION 32 + x
y =
( 0, ) , 13
(0, +3) ,

9. GUIDED PRACTICE
for Examples 1 and 2
Write an equation of the hyperbola with the given foci and vertices.
4. Foci: (–3, 0), (3, 0)
Vertices: (–1, 0), (1, 0) x2 –
y2 8
= 1
SOLUTION
5. Foci: (0, – 10), (0, 10)
Vertices: (0, – 6), (0, 6)
= 1 y2 36 –
x2 64
SOLUTION