Intermediate Algebra (11th edition)

Graphing a Semicircle

Graph Give the domain and range.

Given function

Replace with y.

Square each side.

Add

This is the graph of a circle with center at

and radius 5. Since or y, represents a princi-

pal square root in the original equation, must

be nonnegative. This restricts the graph to the up-

per half of the circle, as shown in FIGURE 24. The

domain is 3 - 5, 5 4 ,and the range is 3 0, 5 4.

ƒ 1 x 2

ƒ 1 x 2 ,

1 0, 0 2

x^2 + y^2 = 25 x^2.

y^2 = 25 - x^2

y= 225 - x^2 ƒ 1 x 2

ƒ 1 x 2 = 225 - x^2

ƒ 1 x 2 = 225 - x^2.

EXAMPLE 4

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

Generalized Square Root Function

For an algebraic expression in xdefined by u, with a function of the form

is a generalized square root function.

ƒ 1 x 2  2 u

uÚ0,

x

y

–5^05

5

f(x) = √25 – x^2

FIGURE 24

NOW TRY

EXERCISE 4

Graph.
Give the domain and range.

ƒ 1 x 2 = 264 - x^2

NOW TRY

EXERCISE 5

Graph.

Give the domain and range.

y

4

=-

B

1 -

x 2

9

NOW TRY ANSWERS
4.

domain: ; range:
5.

domain: ; range: 3 - 3, 3 4 3 - 4, 0 4

3 - 8, 8 4 3 0, 8 4

x

y

–8^028

8

2

f(x) = √64 – x^2

x

y

0

–3

–4

3

9

x^2

4

y= – 1 –

NOTE Root functions like those graphed in FIGURES 24 and 25, can be entered and

graphed directly with a graphing calculator.

NOW TRY

NOW TRY

A 2 a B^2 =a

Graphing a Portion of an Ellipse

Graph Give the domain and range.

Square each side to get an equation whose form is known.

Square each side.

Apply the exponents.

Add

This is the equation of an ellipse with x-intercepts

and and y-intercepts and

Since equals a negative square root in

the original equation, y must be nonpositive, re-

stricting the graph to the lower half of the ellipse,

as shown in FIGURE 25. The domain is and

the range is 3 - 6, 0 4.

3 - 4, 4 4 ,

y

1 0, - 62. 6

1 4, 0 2 1 - 4, 0 2 1 0, 6 2

x 2

16.

x^2

16

+

y^2

36

= 1

y^2

36

= 1 -

x^2

16

a

y

6

b

2

= a-

B

1 -

x^2

16

b

2

y

6

=-

B

1 -

x^2

16

.

EXAMPLE 5

x

y

–6

–4 0 4

y

6

x^2

16

= – 1 –

FIGURE 25