chapter 1:linear functions, equations, and inequalities


Chapter 1: Linear Functions, Equations, and Inequalities

1.1 Real Numbers and the Rectangular Coordinate System

1.2 Introduction to Relations and Functions1.3 Linear Functions1.4 Equations of Lines and Linear Models1.5 Linear Equations and Inequalities1.6 Applications of Linear Functions


1.3 Linear Functions

• Linear Function– a and b are real numbers– Its graph is called a line– Its solution is an ordered pair, (x,y), that makes the equation true

Example

The points (0,6) and (–1,3) are solutions of since 6 = 3(0) + 6 and 3 = 3(–1) + 6.

baxxf )(

63)( xxf63 xy


1.3 Graphing a Line Using Points

• Graphing the line 63 xy

x y

2 0

1 3

0 6

1 9

Plot the ordered pairs

Connect with a straight line.


1.3 Graphing a Line with the TI-83

• Graph the line with the TI-83

Xmin=-10, Xmax=10, Xscl=1 Ymin=-10, Ymax=10,Yscl=1

63 xy


1.3 The x- and y-Intercepts, Zero of a Function

• x-intercept: let y = 0 and solve for x

• y-intercept: let x = 0 and solve for y

• Zero of a function is any number c where f (c) = 0

• Two distinct points determine a line- e.g. (0,6) and (–2,0) are the y- and x-intercepts of the

line y = 3x + 6, and x = –2 is the zero of the function.


1.3 Graphing a Line Using the Intercepts

Example: Graph the line .52 xy

x y

x-intercept 0 5

y-intercept 2.5 0


1.3 Application of Linear Functions

A 100 gallon tank is initially full of water and being drained at a rate of 5 gallons per minute.a) What is the linear function that models this problem?

b) How much water is in the tank after 4 minutes?

c) Interpret the x- and y-intercepts.

1005)(amount initialchange) of rateconstant ()(

xxfxxf

gallons 80100)4(55)4( f

-intercept, let 0 5(0) 100 100meaning that the tank initially has 100 gallons in it.-intercept, let 0 0 5 100 20 minutes

meaning that the tank takes 20 minutes to empty.

y x y

x y x x


1.3 Constant Function

• Constant Function – b is a real number– the graph is a horizontal line– y-intercept: (0,b)– domain– range – Example:

bxf )(

),( }{b

3)( xf


1.3 Graphing with the TI-83

• Different views with the TI-83

• Comprehensive graph shows all intercepts

63)( xxf 63)( xxf


1.3 Slope

• Slope of a Line

$20,082 $5991 $14,091 $705.2004 1984 20

In 1984, the average annual cost for tuition and fees at private four-year colleges was $5991. By 2004, this cost had increased to $20,082. The line graphed to the right is actually somewhat misleading, since it indicates that the increase in cost was the same from year to year.

The average yearly cost was $705.


• Slope m

x

y

0

),(11

yx

),(22

yx

12xxx

12yyy

1.3 Formula for Slope

12

12

xxyy

xym

)1,2( yx


1.3 Example: Finding Slope Given Points

74

74

25)1(3

12

12

xxyym

Determine the slope of a line passing through points (2, 1) and (5, 3).


1.3 Graph a Line Using Slope and a Point

• Example using the slope and a point to graph a line– Graph the line that passes through (2,1) with slope

34

x

y

0

(2,1)

down 4

x

y

0

(2,1)

right 3

(5,-3)

x

y

0

(2,1)

(5,-3)


1.3 Slope of Horizontal and Vertical Lines

• Slope of a horizontal line is 0

• Slope of a vertical line

• Equation of a vertical line that passes through the point (a,b):

(0,4) x

y

0

(1,4)

040

0144

m

x

y

0

(4,4)

4

undefinedm

04

4404

ax


1.3 Slope-Intercept Form of a Line

Slope-intercept form of the equation of a line

– is the slope, and– b is the y-intercept

baxy

baxxf

or

)(

am


1.3 Matching Examples

Solution:

32 .1 xy 32 2. xy 32 .3 xyA. B. C.

1) C, 2) A, 3)B


1.3 Application of Slope

• Interpreting Slope– In 1980, passengers traveled a total of 4.5 billion miles on

Amtrak, and in 2000 they traveled 5.5 billion miles.

a) Find the slope m of the line passing through the points (1980, 4.5) and (2000, 5.5).

Solution:

b) Interpret the slope.Solution:

5.5 4.5 1 0.052000 1980 20m

Average number of miles people are traveling on Amtrak increased by around .05 billion, or 50 million miles per year.

chapter 1:linear functions, equations, and inequalities

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