copyright © 2006 brooks/cole, a division of thomson learning, inc. preliminaries 1 precalculus...
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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Preliminaries
1• Precalculus Review I
• Precalculus Review II
• The Cartesian Coordinate System
• Straight Lines
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
y-axis
x-axis
(x, y)
x
y
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Horizontal Lines
y = 2
Can be expressed in the form y = b
x
y
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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