Intermediate Algebra (11th edition)

11.1 Additional Graphs of Functions

Other Functions
In addition to the squaring function, some other elementary
functions include the following:

• Absolute value function, defined by


• Reciprocal function, defined by


• Square root function, defined by


• Greatest integer function, defined by
  which is a step function.

Their graphs can be translated, as shown in the first three
examples at the right.

ƒ 1 x 2 x,

ƒ 1 x 2  2 x

ƒ 1 x 2 ^1 x

ƒ 1 x 2 x

2

2

4

–4 4

–4

f(x)=[[x]]

y

2 6 x

3

0

y

x

f(x)=x–2+1

2

–2

2

0

y

x

f(x)=x+1^1

–2 2

–2

0

f(x)=x 2

y

x

QUICK REVIEW

CONCEPTS EXAMPLES

11.2 The Circle and the Ellipse

Circle
The circle with radius rand center at has an
equation of the form

Ellipse
The ellipse whose x-intercepts are and and
whose y-intercepts are and has an
equation of the form

x^2

a^2



y^2

b^2

1.

1 0,b 2 1 0,-b 2

1 a, 0 2 1 - a, 0 2

1 xh 22  1 yk 22 r^2.

1 h,k 2 The circle with equation , which can be

written has center and

radius 5.

Graph

0 x

y

–3 3

2

–2

+= 1y 4

2

9

x^2

x^2

9

+

y^2

4

=1.

x

y

–7^0 –2 3

8

(–2, 3)

5

(x+ 2)^2 + ( y– 3)^2 = 25

3 x- 1 - 2242 + 1 y- 322 = 52 , 1 - 2 , 32

1 x+ 222 + 1 y- 322 = 25

11.3 The Hyperbola and Functions Defined

by Radicals

Hyperbola
A hyperbola with x-intercepts and has an
equation of the form

and a hyperbola with y-intercepts and has
an equation of the form

The extended diagonals of the fundamental rectangle
with vertices at the points
and 1 a,-b 2 are the asymptotes of these hyperbolas.

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

y^2

b^2



x^2

a^2

1.

1 0,b 2 1 0,-b 2

x^2

a^2



y^2

b^2

1,

1 a, 0 2 1 - a, 0 2 Graph

The graph has x-intercepts and

The fundamental rectangle has vertices at

and 1 2,- 22.

1 2, 2 2 , 1 - 2, 2 2 , 1 - 2,- 22 ,

1 - 2, 0 2.

1 2, 0 2

x^2

4

-

y^2

4

=1.

x

y

2

2

–2

–2

x 4 – = 1

2

4

y^2

0

670 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems

(continued)

x^2

9 

y^2

4 ^1