Intermediate Algebra (11th edition)

For Exercises 55 – 57, see Section 3.1.

55.Plot the points

56.Sketch the graphs of on the same axes.

57.Find the x- and y-intercepts of the graph of

58.Solve the equation x^2 =121.See Section 9.1.

4 x+ 3 y=12.

y=^43 x and y=-^43 x

1 3, 4 2 , 1 - 3, 4 2 , 1 3, - 42 , and 1 - 3, - 42.

PREVIEW EXERCISES

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

OBJECTIVES OBJECTIVE 1 Recognize the equation of a hyperbola. A hyperbolais the

set of all points in a plane such that the absolute value of the differenceof the dis-

tances from two fixed points (the foci) is constant.

The graph of a hyperbola has two parts, called

branches,and two intercepts (or vertices) that lie on

its axis, called the transverse axis.The hyperbola in

FIGURE 20has a horizontal transverse axis, with foci

and and x-intercepts and

(A hyperbola with vertical transverse axis would have

its intercepts on the y-axis.)

A hyperbola centered at the origin has one of the

following equations. It is shown in more advanced

courses that for a hyperbola, c^2 =a^2 +b^2.

1 c, 0 2 1 - c, 0 2 1 a, 0 2 1 - a, 0 2.

The Hyperbola and Functions Defined by Radicals

11.3

1 Recognize the

equation of a

hyperbola.

2 Graph hyperbolas

by using

asymptotes.

3 Identify conic

sections by their

equations.

4 Graph certain

square root

functions.

x

y

(–c, 0)

Transverse

axis

(–a, 0) (a, 0) (c, 0)

Focus Focus

FIGURE 20

If we were to throw two stones into a pond, the ensuing concentric ripples would

be shaped like a hyperbola. A cross-section of the cooling towers for a nuclear power

plant is hyperbolic, as shown in the photo.

OBJECTIVE 2 Graph hyperbolas by using asymptotes.The two branches of

the graph of a hyperbola approach a pair of intersecting straight lines, which are its

asymptotes. See FIGURE 21on the next page. The asymptotes are useful for sketching

the graph of the hyperbola.

Equations of Hyperbolas

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

Transverse axis on x-axis

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

Transverse axis on y-axis

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

Asymptotes of Hyperbolas

The extended diagonals of the rectangle with vertices (corners) at the points

and are the asymptotesof the hyperbolas

and

y^2

b^2

-

x^2

a^2

=1.

x^2

a^2

-

y^2

b^2

= 1

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