copyright © 2006 brooks/cole, a division of thomson learning, inc. preliminaries 1 precalculus...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Preliminaries

1• Precalculus Review I

• Precalculus Review II

• The Cartesian Coordinate System

• Straight Lines

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The Real Numbers

The real numbers can be ordered and represented in order on a number line

-3 -2 -1 0 1 2 3 4

-1.87

0

4.552

x

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Inequality Graph Interval

3 7x

5x

1

3x

3,7

5,

1,

3

]

( ]

(5

3 7

1

3

) or ( means not included in the solution

] or [ means included in the solution

Inequalities, graphs, and notations

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IntervalsInterval Graph

( )

[ ]

( ]

[ )

(

)

[

]

a b

Example

(a, b)

[a, b]

(a, b]

[a, b)

(a, )

(- , b)

[a, )

(- , b]

(3, 5)

[4, 7]

(-1, 3]

[-2, 0)

(1, )

(- , 2)

[0, )

(- , -3]

( )

[ ]

( ]

[ )

(

)

[

]

a b

a b

a b

a

a

b

b

3 5

-2 0

4 7

-1 3

-3

2

1

0

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Properties of InequalitiesIf a, b, and c are any real numbers, then

Property 1

Property 2

Property 3

Property 4

If ,

then .

a b

a c b c

If and ,

then .

a b b c

a c

If and 0,

then .

a b c

ac bc

If and 0,

then .

a b c

ab bc

Example

2 < 3 and 3 < 8, so 2 < 8.

5 3, so 5 2 3 2;

that is, 3 1.

5 3, and 2 0, we have

( 5)(2) ( 3)(2); that is, 10 6.

5 4, and 2 0, we have

( 5)( 2) (4)( 2); that is,10 8.

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Absolute Value

if 0

if 0

a aa

a a

To evaluate:

3 8

5 2 5 3 5 2 (3 5) 2 5 5

( 5) 5 5Notice the opposite sign

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Absolute Value Properties If a and b are any real numbers, then

Property 5

Property 6

Property 7

Property 8

ab a b

a a

b 0aa

b b

a b a b

Example

2 3 6 2 3

22 2

3 3 3

8 ( 5) 3 8 5 13

4 4 4 4

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Exponents

na 35 5 5 5 125 ...na a a a a

Definition

n factors

Examplen,m positive integers

0a

na

0 1 0a a

10n

na a

a

032 1

44

1 12

162

/m na

/m na

/ nm n ma a

/ 1m n

n ma

a

32 / 3 2125 125 25

3/ 2 34 9 27

9 4 8

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Laws of Exponents

m n m na a a

Law Example

nm mna am

m nn

aa

a

n n nab a bn n

n

a a

b b

3 12 3 12 15x x x x

65 5(6) 303 3 3 14

14 12 212

yy y

y

4 4 4 43 3 81r r r 3 3

3 3

4 4 64

x x x

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Algebraic Expressions

• Polynomials

• Rational Expressions

• Other Algebraic Fractions

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Polynomials• Addition

3 2 33 2 7 15 5 13 12x x x x x 3 2 3

3 2

3 2 7 15 5 13 12

8 2 6 27

x x x x x

x x x

Combine like terms

• Subtraction

3 2 3 26 1 3 2x x x x x x 3 2 3 2

3

6 1 3 2

2 4 1

x x x x x x

x x

Combine like terms

Distribute

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Polynomials• Multiplication

2 5 3 2x x

Combine like terms

Distribute2 (3 2) 5(3 2)x x x

Distribute26 4 15 10x x x 26 11 10x x

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Factoring Polynomials

3 26 36t t

• Greatest Common Factor

• Grouping

26 6t t

2 2 2mx mx x

1 2 1mx x x

The terms have 6t2 in common

2 1mx x

Factor mx Factor –2

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Factoring Polynomials

• Sum/Difference of Two Cubes:

• Difference of Two Squares:

2 9m

38 1x 22 1 4 2 1x x x

3 3m m

2 2x y x y x y

3 3 2 2x y x y x xy y

Ex.

Ex.

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Factoring Polynomials• Trinomials

2 5 6x x

3 26 27 12x x x

3 2x x

Ex.

Ex.

Trial and Error

23 2 9 4x x x

Trial and Error 3 2 1 4x x x

Greatest Common Factor

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Roots of Polynomials• Finding roots by factoring

2 1 3 0

2 1 0 or 3 0

1 or 3

2

x x

x x

x x

22 5 3 0x x

(find where the polynomial = 0)

Ex.

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Roots of Polynomials

• The Quadratic Formula:

If 2 0 0ax bx c a

2 4

2

b b acx

a

• Finding roots by the Quadratic Formula

with a, b, and c real numbers, then

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ExampleUsing the Quadratic Formula:

27 7 4 3 1 7 37

2 3 6x

Ex. Find the roots of 23 7 1 0x x

Here a = 3, b = 7, and c = 1

Plug in

Note values

7 37 7 37 or

6 6x

Simplify

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Rational ExpressionsP, Q, R, and S are polynomials

Addition

Operation

Multiplication

Subtraction

Division

P Q P Q

R R R

P Q P Q

R R R

P Q PQ

R S RS

P Q P S PS

R S R Q RQ

Notice the common denominator

Find the reciprocal and multiply

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Rational Expressions• Simplifying

2

2

25

7 10

x

x x

5 5

2 5

x x

x x

Cancel common factorsFactor

• Multiplying

2 2

3 2

2 1 6 6

1

x x x x

x x

3

1 1 6 1

1 1

x x x x

x xx

FactorCancel common factors

2

Multiply Across

5

2

x

x

2

6 1x

x

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Rational Expressions• Adding/Subtracting

3 2

4x x

Combine like terms

3 4 2

( 4) 4

x x

x x x x

Must have LCD: x(x + 4)

3 12 2 5 12

( 4) 4

x x x

x x x x

Distribute and combine fractions

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Other Algebraic Fractions• Complex Fractions

32

94

x

xx

Factor to get hereDistribute and reduce to get here

32

94

xx

x xx

2

3 2

9 4

x

x

Multiply by the LCD: x

3 2 1

3 2 3 2 3 2

x

x x x

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Other Algebraic Fractions

• Rationalizing a Denominator

7

3 y

Simplify

7 3

3 3

y

y y

21 7

9

y

y

Multiply by the conjugate

Notice: a b a b a b

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Cartesian Coordinate System

y-axis

x-axis

(x, y)

x

y

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Cartesian Coordinate System

x

Ex. Plot (4, 2)

(4, 2)

Ex. Plot (-2, -1)

Ex. Plot (2, -3)

(2, -3)(-2, -1)

y

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The Distance Formula

y

2 2,x y

1 1,x y

2 21 2 1 2d x x y y

x

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The Distance Formula

y

7,5

3, 2

2 21 2 1 2

2 27 ( 3) 5 ( 2)

100 49 149

d x x y y

d

d

Ex. Find the distance between (7, 5) and (-3, -2)

7

10

x

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The Equation of a Circle

2 2 2x h y k r

A circle with center (h, k) and radius of length r can be expressed in the form:

Ex. Find an equation of the circle with center at (4, 0) and radius of length 3

2 2 2

2 2

4 0 3

4 9

x y

x y

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Straight Lines

• Slope

• Point-Slope Form

• Slope-Intercept Form

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Slope – the slope of a non-vertical line that passes through the points

is given by:

and

2 1

2 1

y yym

x x x

Ex. Find the slope of the line that passes through the points (4,0) and (6, -3)

3 0 3 3

6 4 2 2

ym

x

1 1,x y 2 2,x y

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Slope

Two lines are parallel if and only if their slopes are equal or both undefined

Two lines are perpendicular if and only if the product of their slopes is –1. That is, one slope is the negative reciprocal of the other slope (ex. ).3 4

and 4 3

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Point-Slope Form

1 1

1 4 3

1 4 12

4 11

y y m x x

y x

y x

y x

An equation of a line that passes through the point with slope m is given by:

1 1,x y

Ex. Find an equation of the line that passes through (3,1) and has slope m = 4.

1 1y y m x x

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Slope-Intercept Form

44

5

y mx b

y x

An equation of a line with slope m and y-intercept is given by: 0,b

Ex. Find an equation of the line that passes through (0,-4)

and has slope .

y mx b

4

5m

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Vertical Lines

x = 3

Can be expressed in the form x = a

x

y

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Horizontal Lines

y = 2

Can be expressed in the form y = b

x

y

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Example

Since the slope of the line 2 7 is 2,

1we have the slope of the perpendicular line is .

2

y x

m

Find an equation of the line that passes through (-2, 1) and is perpendicular to the line

Solution:

Step 1.

2 7.y x

Step 2. 1 11

1 22

1 11 1

2 2

y y m x x y x

y x y x

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Example

We need to find the slope of the line 6 2 5.

52 6 5 3 3.

2The slope of the parallel line is also 3.

x y

y x y x m

m

Find an equation of the line that passes through (0, 1) and is parallel to the lineSolution:

Step 1.

6 2 5.x y

Step 2. Since 3 and 1,

3 1

m b

y mx b y x