Chapter 2

Linear Functions and Models

Ch 2.1 Functions and Their Representations

A function is a set of ordered pairs (x, y), where

each x-value corresponds to exactly one y-value.

 

Input x Output yFunction f

(x, y)

Input Output

…continued

y is a function of x because the output y is determined by anddepends on the input x. As a result, y is called the dependent variableand x is the independent variable To emphasize that y is a function of x, we use the notation y = f(x)

and is called a function notation. y = f(x) Output Input 14 = f(5)

  A function f forms a relation between inputs x and outputs y that canbe represented verbally (Words) , numerically (Table of values) ,Symbolically (Formula), and graphically (Graph).  

y x

Representation of Function

x (yards

1 2 3 4 5 6 7

y(feet)

3 6 9 12 15 18 21

0 4 8 12 16 20 24

20

16

12

8

4

y= 3x

Table of Values Graph

x

y

Numerically Graphically

Diagrammatic Representation (pg 76)

Function Not a function

1

2

3

3

6

91

2

4

5

6

(1, 3), (2, 6), (3, 9)

x y x y

1

2

3

4

5

(1,4), (2, 4), (3, 5)(1, 4), (2, 5), (2, 6)

Domain and Range Graphically (Pg 80)

-3 -2 -1 0 1 2 3

Domain

Range

The domain of f is the set of all x- values,and the range of f is the set of all y-values

3

2

1

Range R includes all y – values satisfying 0 < y < 3

x

Domain D includes all x valuesSatisfying –3 < x < 3

y

Vertical Line Test ( pg 83)

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

5

4

3

2

1

-1

-2

-3

-4

-5

(-1, 1)

(-1, -1)

If each vertical line intersects the graph at most once, then it is a graph of a function

Not a function

…Continued

-3 -2 -1 0 1 2 3

(-1, 1)

(1, -1)

4

3

2

1

-1

-2

-3

Not a function

Using Technology

[ - 10, 10, 1] by [ - 10, 10, 1]

Hit Y and enter 2x - 1

x y

-1 -3

0 -1

1 1

2 3

Graph of y = 2x - 1

Hit 2nd and hit table and enter data

2.2 Linear Function

A function f represented by f(x) = ax + b, where a and b are constants, is a linear function.

0 1 2 3 4 5 6 0 1 2 3 4 5 6

100

90

80

70

60

100

90

80

70

60

Scatter Plot A Linear Function

f(x) = 2x + 80

Modeling data with Linear Functions Pg ( 97)

Example 7 1500

1250

1000

750

500

250

0

4 8 12 16 20 x

Credits

Cost (dollars)

Symbolic Representation f(x) = 80x + 50

Numerical representation 4 8 12 16 $ 370 $ 690 $1010 $1330

Using a graphing calculator

Example 5 (pg 95)

Give a numerical and graphical representation

f(x) = 1 x - 2

2

Numerical representation

Y1 = .5x – 2 starting x = -3

Graphical representation [ -10, 10, 1] by [-10, 10, 1]

2.3 The Slope of a line

1 2 3 4 5 6 x

Gasoline (gallons )

Cost of GasolineEvery 2 gallons purchased the cost increases by $3

8

7

6

5

4

3

2

1Run = 2

Rise = 3

Slope = Rise = 3 Run 2

Y

Cost(dollars)

2.3 Slope (Pg 106)

The Slope m of the line passing through the points (x1 y1 ) and (x2, y2) is

m= y2 –y1/ x2 –x1

Where x1 = x2. That is, slope equals rise over run.  

y2 (x2, y2)

y2 –y1

y1 (x1, y1) x2 –x1

rise y2 - y1

m = run = x2

- x1

 

Run

Rise

m = - ½ < 0m = 2 > 0

m = 0

m is undefined

Positive slope

Negative slope

Zero slopeUndefined slope

-4 -2 1 2 3 4

4

3

2

1

-1

-2

-3

-1

2

4

3

2

1

0

-1

-2

- 4 -2 1 2

2

(Pg 107)

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

4

3

2

1

-1

- 2

-3

- 4

( 3, 2)

(0, 4)

Example 2 - Sketch a line passing through the point (0, 4) and having slope - 2/3

y - valuesdecrease 2 units each times x- values increase by 3(0 + 3, 4 – 2)= (3, 2)

( 0, 4)

Rise = -2

Slope-Intercept Form ( pg 109) 

The line with slope m and y = intercept b is given by

 

y= mx + b

 

The slope- Intercept form of a line

Example – 4 (pg 109)

-3 -2 -1 1 2

y = ½ x + 2

y = ½ x

y = ½ x - 2

3

2

1

-1

-2

-3

Analyzing Growth in Walmart

Example 10

0 1999 2003 2007

3.0

2.5

2.0

1.5

1.0

0.5

m1 = 1.1 – 0.7 = 0.2 m2 = 1.4 - 1.1 = 0.1 and 1999 – 1997 2002 – 1999 m3 = 2.2 - 1.4 = 0.16 2007 - 2002

Years

Employees(millions)

Year 1997 1999 2002 2007

Employees 0.7 1.1 1.4 2.2

m1

m2

m3

Average increase rate

2.4 Point- slope form ( pg 119)

The line with slope m passing through the point (x1 , y1 ) is given by

y = m ( x - x1 )+ y1

Or equivalently,

y – y1 = m (x –x1)

The point- slope form of a line

(x1, y1)

(x, y)

x – x1

y – y1

m =( y – y1) / (x – x1)

Horizontal and Vertical Lines (pg 125)

x = h

b

h

y= b

Equation of Horizontal Line Equation of vertical line

x

y

x

y

…Continued(Pg 126 – 127)

Parallel Lines

Two lines with the same slope are parallel.

m1 = m2

Perpendicular Lines

Two lines with nonzero slopes m1 and m2 are

perpendicular

if m1 m2 = -1

Pg 127

m2 = -1 m2 = - 1/2 m2 = - 1/m1m1 = 1m1 = 2 m1

Perpendicular Lines