Intermediate Algebra (11th edition)

SECTION 11.3 The Hyperbola and Functions Defined by Radicals 651

This rectangle is called the fundamental rectangle.Using the methods of Chapter 3,

we could show that the equations of these asymptotes are

and

Equations of the

asymptotes of a hyperbola

To graph hyperbolas, follow these steps.

y

b

a

y x.

b

a

x

Graphing a Hyperbola

Step 1 Find the intercepts.Locate the intercepts at and if

the has a positive coefficient, or at and if the

-term has a positive coefficient.

Step 2 Find the fundamental rectangle.Locate the vertices of the funda-

mental rectangle at and

Step 3 Sketch the asymptotes.The extended diagonals of the rectangle are

the asymptotes of the hyperbola, and they have equations

Step 4 Draw the graph.Sketch each branch of the hyperbola through an

intercept and approaching (but not touching) the asymptotes.

y= ba x.

1 a, b 2 , 1 - a, b 2 , 1 - a, -b 2 , 1 a, -b 2.

y^2

x^2 -term 1 0, b 2 1 0, -b 2

1 a, 0 2 1 - a, 0 2

Graphing a Horizontal Hyperbola

Graph

Step 1 Here and The x-intercepts are and

Step 2 The four points and are the vertices of

the fundamental rectangle, as shown in FIGURE 21below.

Steps 3 The equations of the asymptotes are and the hyperbolaapproaches

and 4 these lines as xand yget larger and larger in absolute value.

y= 

5

4 x,

1 4, 5 2 , 1 - 4, 5 2 , 1 - 4, - 52 , 1 4, - 52

a= 4 b= 5. 1 4, 0 2 1 - 4, 0 2.

x^2

16

-

y^2

25

=1.

EXAMPLE 1

4

x

y

(– 4 , – 5 ) ( 4 , , – 5 )

(– 4 , 5 ) ( 4 , , 5 )

• 4

(^3) xx 22
1616
yy^22

• 2525 11

0

FIGURE 21

Be sure that

the branches

do not touch

the asymptotes.

NOW TRY

NOW TRY

EXERCISE 1

Graph.

x^2

25

-

y^2

9

= 1

NOW TRY

EXERCISE 2

Graph

y^2

9

-

x^2

16

=1.

x

y

–3

–5 5

3

0

–= 1y 9

2

25

x^2

x

y

–3

–4 4

3

0

–= 1 16 x

2

9

y^2

4

x

y

(– 4 , – 7 )

(– 4 , 7 )

( 4 , – 7 )

( 4 , 7 )

• 4
  
  
  • 7
  
  
  
  

7

y^2

49

x^2

• 16 1
  0

FIGURE 22

Graphing a Vertical Hyperbola

Graph

This hyperbola has y-intercepts and The asymptotes are the

extended diagonals of the rectangle with vertices at and

1 4, - 72 .Their equations are y= 74 x.See FIGURE 22above. NOW TRY

1 4, 7 2 , 1 - 4, 7 2 , 1 - 4, - 72 ,

1 0, 7 2 1 0, - 72.

y^2

49

-

x^2

16

=1.

EXAMPLE 2

NOW TRY ANSWERS

1. 
   
   
   
   2.