Intermediate Algebra (11th edition)

9.4 Formulas and Further Applications

To solve a formula for a squared variable, proceed as
follows.

(a) If the variable appears only to the second power:
Isolate the squared variable on one side of the equation,
and then use the square root property.

(b) If the variable appears to the first and second
powers:Write the equation in standard form, and then
use the quadratic formula.

Solve for r.

Multiply by

Divide by A.

Square root

property

Rationalize

r= denominator.

 22 mpA

A

r=

B

2 mp

A

r^2 =

2 mp

A

r^2 A= 2 mp r^2.

A=

2 mp

r^2

CONCEPTS EXAMPLES

Solve for x.

Standard

form

x=

• r 2 r^2 + 4 t
  2

a=1,b=r,c=-t

x=

• r 2 r^2 - 41121 - t 2
  2112

x^2 +rx-t= 0

x^2 +rx=t

9.5 Graphs of Quadratic Functions

1.The graph of the quadratic function defined by
, is a parabola with
vertex at and the vertical line as axis.

2.The graph opens up if ais positive and down if ais
negative.

3.The graph is wider than the graph of if
06 |a| 61 and narrower if |a| 7 1.

ƒ 1 x 2 =x^2

1 h, k 2 x=h

F 1 x 2 a 1 xh 22 k,aZ 0

Graph

The graph opens down since

Vertex:

Axis:

Domain:

Range: 1 - q, 1 4

1 - q, q 2

x=- 3

1 - 3, 1 2

a 6 0.

x

y

0

1

–2

–5 1

(–3, 1)

x = –3

ƒ 1 x 2 =- 1 x+ 322 +1.

9.6 More about Parabolas and Their

Applications

The vertex of the graph of
may be found by completing the square.

The vertex has coordinates

Graphing a Quadratic Function

Step 1 Determine whether the graph opens up or
down.

Step 2 Find the vertex.

Step 3 Find the x-intercepts (if any). Find the y-
intercept.

Step 4 Find and plot additional points as needed.

Horizontal Parabolas

The graph of

or

is a horizontal parabola with vertex and the
horizontal line as axis. The graph opens to the right
if and to the left if

Horizontal parabolas do not represent functions.

a 70 a 6 0.

y=k

1 h, k 2

xay^2 byc xa 1 yk 22 h

A 2 ab , ƒA 2 abBB.

aZ0,

ƒ 1 x 2 ax^2 bxc, Graph

The graph opens up since

Vertex:

The solutions of are

and so the x-intercepts are

and

so the y-intercept is

Domain:

Range:

Graph

The graph opens to the right

since

Vertex:

Axis:

Domain:

Range: 1 - q, q 2

C^12 , qB

y=-^32

A^12 , -^32 B

a 7 0.

x

y

0

(^11)
–1^5
1
2
3
(), – 2
3
y = – 2
x= 2 y^2 + 6 y+5.
3 - 1, q 2
1 - q, q 2
ƒ 102 =3, 1 0, 3 2.

1 - 3, 0 2.

- 3, 1 - 1, 0 2

x^2 + 4 x+ 3 = 0 - 1

1 - 2, - 12

a 7 0.

x

y

0

3

–3 1

(–2, –1)

x = –2

–1

ƒ 1 x 2 =x^2 + 4 x+3.

CHAPTER 9 Summary 561

(continued)