Linear Models

Year Population(millions)

2003 8.41

2004 8.52

2005 8.66

2006 8.85

2007 9.04

2008 9.22

Create a scatterplot of the population for North Carolina on your graphing calculator.

Linear Models

Define the variables

t = years since 2000

P = population of North Carolina in millions

Linear Models: TI –84

Input data: Go to STAT and 1:Edit Go to the top of L1 and clear Go to the top of L2 and clear

L1 represents x or horizontal axisL2 represents y or vertical axis

Go to L1 and enter the valuesGo to L2 and enter the values on the

table

Linear Models: TI –84

Domain/RangeSelect WINDOW and you will see:Xmin = smaller than smallest inputXmax= larger than largest inputXscal= Ymin= smaller than the smallest outputYmax= larger than largest outputYscal=Each will be based on the domain and range of

the values from the table.

Linear Models: TI –84Graphing Points

• Enter 2nd and y =• A screen with STAT PLOTS• Choose 1:1: Plot 1 enter• Make sure Plot 1 is on and the X list is

designated L1• Y list: L2• Also make sure Type: shows scatter plot.• Finally select Graph and the plot will show on

the screen.

Graphing Line on TI- 84

• Choose the two points that will make the line the best fit, and solve for the equation

• 8.41 – 9.04 3 - 7

m = 0.1575 (round to nearest hundredth)Y-intercept y = 0.16x + b 8.41 = 0.16(3) + b 8.41 = 0.48 + b 7.93 = b

Graphing Line on TI- 84

y = 0.16x + 7.93Select y = , buttonEnter the equation found 0.16x + 7.93Use the X,T,ϴ,n button to insert x into the

equation.Once the equation is entered select GRAPHA line will be drawn through the points that

were previously plotted.

Total Revenue for GE

a) Find an equation for a model of these data.

b) Using your model, estimate GE’s revenue in 2010

c)What is the slope of your model?

What does it mean in regard to GE’s revenue

Year Revenue

(Billions $)

2004 124

2005 136

2006 152

2007 172

2008 183

Relation

Relationship between elements of a set of input and elements of a set of outputs

(2,3)

x + 2y = 12

Relates x-values with y-values

using arithmetic operation

Function

For every input there is only one output

{(1,2) , (3,4) , (5, 6) , (7,8)}

123

567

Input Output x y

1 5

2 5

3 5

Function or Not?

• The set A = {(2,5) , (4,8) , (10,8) , (20, 15)}

• Weekly salaries during the mth month of the year.

Day of week Monday Wednesday Saturday Monday

Temperature

F

90 88 91 93

Notation

f(x)

“f of x” Represents a function named f that

depends on the variable xf means output/y-variable/range

Shorthand method of providing information in a compact form.

Word problems and Function Notation

a) H(t) = height of a toy rocket in feet t seconds after launch.

What does H(3) = 12 mean?

b) C(m) = Cost in hundred of dollars for producing m miracle mops.

C (2500) = 189

c) P(t) = population of Michigan, in millions, t, years since 2000.

P(10) = 10.4

Population of Wisconsin, in millions

• Find an equation for a model of these data. Write your model in function notation.

• Determine a reasonable domain and range

• Find P(14) and interpret its meaning in regard to the pop of Wisconsin.

Year Population

(millions)

2003 5.47

2004 5.51

2005 5.54

2006 5.57

2007 5.60

2008 5.63

Review

• f(x) = 4x + 3 f(3)

Domain and range is not restricted

Word problems Domain and range is set by the problem.

Systems of Equations

Set of equations that require a solution that will work for all of the equations in the set.

y= x + 4y = 2x – 8

• Table

• Graph find the slope and y-intercept for each line

x y = x + 4

2

4

6

x y = 2x - 8

2

4

6

Practice

• Your company wants to print some flyers for advertising a new product. The printer has two options to produce the flyers, The traditional printing cost is $250 for setup and $0.15 per page printed. To print the flyers digitally, they charge $50 for setup and $0.20 per printed page.

Types of Systems

Consistent Inconsistent

Algebraic Methods of solving systems

• Substitution

y = 4x –5

y = x + 22

Replace the expression of a variable in one equation into another.

Substitution

• Geothermal and wind energy, find the year when the amount of geothermal and wind energy produced will be the same.

G(t) = -0.08t + 5.68

W(t) = 0.82t – 1.03