chapter 1:linear functions, equations, and inequalities

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Copyright © 2007 Pearson Education, Inc. Slide 1-2

Chapter 1: Linear Functions, Equations, and Inequalities

1.1 Real Numbers and the Rectangular Coordinate System

1.2 Introduction to Relations and Functions

1.3 Linear Functions

1.4 Equations of Lines and Linear Models

1.5 Linear Equations and Inequalities

1.6 Applications of Linear Functions

Copyright © 2007 Pearson Education, Inc. Slide 1-3

1.1 Real Numbers and the Rectangular Coordinate System

Sets of Real Numbers:• Natural Numbers:• Whole Numbers:• Integers:• Rational Numbers:

• Irrational Numbers:

{0, 1, 2, 3, 4, 5,....}

{1, 2, 3, 4, 5,....}

{..., 3, 2, 1, 0, 1, 2, 3,...}

0 integers, are and , qqpqp

.etc,2, i.e.

decimals, repeatingor fractionsby

drepresente becannot that numbers

Copyright © 2007 Pearson Education, Inc. Slide 1-4

1.1 Example

Indicate the set each number belongs to:

6.0 decimal) (repeating Rational 32

decimal) ng(terminati Rational 41

ating)not termin

decimal repeating,(not Irrational 3.14159

25.0

Copyright © 2007 Pearson Education, Inc. Slide 1-5

1.1 The Set of Real Numbers and the Number Line

• Real Numbers:

• Every real number corresponds to a point on the number line.

sIrrational Rationals

-4 -3 -2 -1 0 1 2 3 4

Copyright © 2007 Pearson Education, Inc. Slide 1-6

1.1 The Rectangular Coordinate System

• The number corresponding to a particular point on the number line is called the coordinate of the point.

• This correspondence is called a coordinate system.

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1.1 The Coordinate Plane

• Cartesian Coordinate System– xy-plane (or coordinate plane)

Quadrant IQuadrant II

Quadrant III Quadrant IV

OriginP(a, b)

Copyright © 2007 Pearson Education, Inc. Slide 1-8

1.1 The TI-83 Viewing Window

• Limitations in portraying coordinate systems on the calculator screen

1. Resolution 2. Scaling

Xmin=-60, Xmax=60, Xscl=1 Xmin=-60, Xmax=60, Xscl=10

Ymin=-40, Ymax=40, Yscl=1 Ymin=-40, Ymax=40,Yscl=10

60

40

40

60

40

40

6060

Copyright © 2007 Pearson Education, Inc. Slide 1-9

1.1 Rounding Numbers

Mode Setting Display

Number Nearest Tenth Nearest Hundredth Nearest Thousandth

1.3782 1.4 1.38 1.378

201.6666 201.7 201.67 201.667

.0819 .1 .08 .082

Copyright © 2007 Pearson Education, Inc. Slide 1-10

1.1 Roots

• Calculators have the ability to express numbers like:

• Other special keys:

3 87 4 12

2 , , x

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1.1 The Distance Formula

• Pythagorean Theorem:

22

222

bac

cba

a

b

c

),(11

yxP

),(22

yxR

x

y

|x2-x1|

|y2-y1|

2

12

2

12)()(),( yyxxRPd

Q (x1, y2)

d

Copyright © 2007 Pearson Education, Inc. Slide 1-12

1.1 Example Using the Distance Formula

• Find the length of the line segment that joins the points P(8, 4) and Q(3, 2).

Solution: 22 )42())8(3(),( QPd

22 )6(11

157

Copyright © 2007 Pearson Education, Inc. Slide 1-13

• The midpoint of the line segment with endpoints and is

Example• Find the midpoint M of the segment with endpoints

(8, 4) and (9,6).

Solution:

1.1 Midpoint Formula

),(11

yx ),(22

yx

1 21 2 ,2 2

y yx x

8 ( 9) 4 6 1 2, , ,12 2 2 2

12

Copyright © 2007 Pearson Education, Inc. Slide 1-14

1.1 Application: Estimating Tuition and Fees

• In 1998, average tuition and fees at public universities and colleges were $3293, whereas they were $5132 in 2004. Use the midpoint formula to estimate tuition and fees in 2001. Compare it to the actual value of $4221.

Notice that 2001 lies midway between 1998 and 2004. Therefore we can use the midpoint formula.

1998 2004 3293 5132, (2001,4212.50)2 2

The midpoint formula estimates tuition and fees at public colleges and universities to be $4212.50 in 2001. This is within $10 of the actual value.