sec 8 - gwinnett county public schools€¦ · sec 8.4 – matrices transformations voting methods...

Sec 8.1 – Matrices Review

Voting methods & Matrix Applications Name:

Use the following matrices for the problems 1 - 13. Show ALL WORK by HAND and check with the calculator.

57

23A

37

25B

621

123C

23

34

02

D 5 3

0 7 1

xE

y

1. 2. TD

3. EC = 4. CD =

5. AB 32 = 6. DC =

7. B D = 8. B E =

What are the dimensions

of Matrix D?

Use the following matrices for the problems 1 - 13. Show ALL WORK by HAND and check with the

calculator.

57

23A

37

25B

621

123C

23

34

02

D 5 3

0 7 1

xE

y

9. C22 + D31 – B21 = 10. 2

A

11. T

D D

12. Create a 2x2, 3x3, and 5x 5 IDENTITY matrices

13. Are Matrices A and B inverses? Demonstrate how you can tell.

14. 2 2 2

2 1 6

a

b

14.

4 1 6 6 27

2 3 2 8

a

b c

15. The determinant of

35

24by hand and show work. 14. Find the determinant of

641

320

153

by hand and show work.

15. Find the inverse of

35

24by hand and show work. 16. Find the inverse of

24

36by hand and show work.

17. On one weekend, the Goxfield Theater reported the following ticket sales for three

first-run movies, as shown in the matrix at the right. If the ticket prices were $6 for

each adult and $4 for each child, what were the weekend sales for each movie?

Determine possible values for a, b, and c that would

make the following matrix statement true.

Determine possible values for a and b that would

make the following matrix statement true.

a =

b =

c =

a =

b =

Sec 8.2 – Informational Matrices


1. A flu epidemic occurs in Middletown schools and each student is either susceptible, sick, or infected.

The percentage of students in each category by grade

level is given below.

The population distribution of the Middletown school district is

given below

Infected

Sick

eSusceptibl

S

%20%10%70

%20%15%65

%15%25%60

HighSchool

olMiddleScho

Elementary

P

11001115

795830

15801610

a. Describe the value of S3 2 and what it represents.

b. Describe the value of P2 2 and what it represents.

c. Find PS . (You can use your calculator if you would like).

INCLUDE LABELS

d. Describe the value of 12PS and what it represents.

e. Based on the flu epidemic in Middletown

schools, how many sick girls are there?

f. Based on the flu epidemic in Middletown

schools, how many boys are infected?

2. A study is being conducted by Time Warner about the network spending of its Network HBO and 2 online competitors.

The percentage of spending categories by each network

is shown below.

The amount of spending per month by each company is shown

below.

Movies

S Mini Series

CurrentTV

60% 35% 70%

30% 30% 30%

10% 35% 0%

NetFlix

P HULU

HBO

$85

$33

$912

million

million

million

a. Describe the value of S2 3 and what it represents.

b. Describe the value of P31and what it represents.

c. Find PS . (You can use your calculator if you would like).

INCLUDE LABELS

d. Describe the value of 1 1S P and what it represents.

e. Based on the study, how much do the 3 companies spend on

Mini Series?

f. Based on the study how much does HBO spend on

Movies and their licensing?

Girls Boys Elementary Middle High

School School School

Monthly Spending NetFlix HBO HULU

3. following matrices show information collected by an apparel store in Magic Kingdom park at Disney World

The matrix [N] shows the number of items sold on

Monday

The matrix [P] shows the Sale Price and Profit of each type of

item.

Adult

Youth

ToddlerInfant

N

/

251824

81530

42375

Hats

Shirts

Jackets

P

$75 $40

$30 $15

$16 $9

a. Describe the value of N23 and what it represents.

b. Describe the value of P2 1 and what it represents.

c. Find PN . (You can use your calculator if you would like).

INCLUDE LABELS

d. Describe the value of 23PN and what it represents.

e. Based on the information what was the total Sales on

Monday for Adult Apparel?

f. Based on the information what was the total

Profit made on Monday on Infant Apparel?

Sale Profit Price Jackets Shirts Hats

Sec 8.3 – Basic Voting Methods Voting methods & Matrix Applications Name:

Consider the following preference schedules for an election.

1. How many preference schedules are possible (if ties are not permitted)?

2. Who is the plurality winner?

What is the percentage of 1st place votes each received?

3. How many first place votes would be needed in this example for there

to be a majority winner?

If there is a majority winner who is it?

4. Who is the ‘run off’ winner?

5. Who is the ‘sequential run off’ winner?

6. What is each candidates Borda count?

Who is the ‘Borda Count’ winner?

7. What is each candidates Condorcet winner?

#5)

#3)

#3)

#6)

Droid:

iP6:

Glxy5:

Z30:

#1)

#4)

#6)

Drd iP6 G5 Z30

Drd *

iP6 *

G5 *

Z30 *

#7)

The BIG QUESTION:

WHO REALLY WINS? SUPPORT YOUR REASONING.

#2)

***Demonstrate how this can be done with matrix multiplication****

iPhone 6 (Apple)

Galaxy 5 (Samsung)

Bold Z30 (Blackberry)

Droid 4 (Motorola)

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

8 5 6 7

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola) iPhone 6

(Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

iPhone 6 (Apple)

Galaxy 5 (Samsung)


Droid 4 (Motorola)

8 5 6 7

Twenty-two Discrete Math students are arguing over which fast food restaurants and listed their preferences below.

1. In your own words, give a description of the plurality winner

What percentage of 1st place votes does each of the following choices have?

Choice Five Guys Chic-fil-a Zaxby’s Wendy’s

Percentage of 1st

place votes

Who is the plurality winner?

2. a.What is the minimum number of first place votes needed in this

example for there to be a majority winner?

b.If there is a majority winner who is it?

3. In your own words, give a description of the ‘run off’ winner :

Who is the ‘run off’ winner?

4. In your own words, give a description of the ‘sequential run off’ winner :

Who is the ‘sequential run off’ winner?

5. In your own words, give a description of the ‘Borda Count’ winner (on a separate page show

how this might be done using Materices):

a. Give the Borda Count for each letter:

b. Who is the ‘Borda Count’ winner?

6. Determine the Condorcet Winner.

#3a)

#3b)

#4)

5)

#6b)

#6a)

5G: C: Zx: W:

→ 5G C Zx W

5G *

C *

Zx *

W *

Chic-fil-a

Zaxby’s

Wendy’s

Five Guys

Wendy’s

Chic-fil-a

Zaxby’s

Five Guys

Chic-fil-a

Zaxby’s

Wendy’s

Five Guys

Wendy’s

Zaxby’s

Chic-fil-a

Five Guys

8 5 6 3

2)

•

•

•

• •

•

Sec 8.4 – Matrices Transformations


Start by creating a picture on a Cartesian coordinate system (preferably

in the first quadrant)

The picture at the right would be represented by the matrix:

1447311

1144641

Enter this into the Matrix [A] in the calculator.

Press MATRX , ◄ , ENTER . Change the

dimensions of the matrix to match the points of

your picture. For our example we will need to

change the dimensions to 2 x 7

To return to the home screen press 2nd

MODE

To have the calculator show your original picture. Press MATRX , ►

, 8 . This will bring up Matr►list( on the calculator.

This function will enable us to put our matrix into the table of the

calculator which can be graphed.

Next, press MATRX , 1 . Before the Matrix can be transformed into

the table it has to be turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd ,

1 , , , 2nd , 2 , ) , ENTER .

Next hit 2nd

, Y= (Stat Plot) , 1.

Make sure your screen has the following options highlighed

Finally push ZOOM, 6.

Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar.

To have the calculator show your original picture. Press MATRX , ► , 8 . This will bring up

Matr►list( on the calculator. This function will enable us to put our matrix into the table of

the calculator which can be graphed.

Next, press, 2 , MATRX , 1 . Before the Matrix can be transformed into the table it has to be

turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd , 3 , , , 2nd

, 4 , ) , ENTER .

Next hit 2nd




Translation To translate an object we have to set up a translation matrix. We can enter this in matrix [B]

5555555

7777777

Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C]

01

10

Fill in all of the points all the way

down and make certain you finish

with the point you

started with (4,7)

to connect the last

line.

Shift left 7 and down 5

Rotation by 90

General Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.)

cos 160 sin 160

sin 160 cos 160

Combination Transformations (Rotate 200° and Translate Right 6) To translate an object we have to set up the rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.) and a

translation matrix in [B].

5 5 5 5 5 5 5

0 0 0 0 0 0 0B

cos 200 sin 200

sin 200 cos 200C

Programmers often use matrices to write visual code even for the mouse cursor For

example let’s suggest that the points A,B,C,D,E,F,G represent a mouse cursor.

Programmers would use the matrix:

a. Draw the original mouse cursor on the graph using a blue colored

pencil.

b. Usually mouse cursors are translated. To do this

programmers add by a translation matrix. Add the

following translation matrix to [S] and draw the new

image using a red pen.

4444444

8888888S

c. To rotate an object counter clockwise programmers

multiply a rotation matrix and [S]. Multiply the

following and graph using a black or gray pencil.

(should be degree mode)

3455232

2325543S

GFEDCBA

S

130cos130sin

130sin130cos

Rotation by 160

Press the MODE key and

switch to DEGREE mode

Rotation by 200

Press the MODE key and

switch to DEGREE mode

Shift right 5

Matrices are used to describe most graphics on computer (including

computer/video games). Consider the following Phoenix bird. As a matrix it

would be described as:

4 4 0 5 5 2 1 3 0 1 4

7 4 7 4 7 0 3 3 4 0 7

A B C D E F G H I J A

Once entered as a matrix several graphical transformations can be performed using matrix operations.

DILATIONS:

To dilate (bigger or smaller) from the origin point, you would only need to multiply the original matrix by a scalar multiple. e. g.

4 4 0 5 5 2 1 3 0 1 42

7 4 7 4 7 0 3 3 4 0 7

……would make the picture twice as big

12

4 4 0 5 5 2 1 3 0 1 4

7 4 7 4 7 0 3 3 4 0 7

……would make the picture shrink the picture to half size

TRANSLATION Matrices:

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

7 4 7 4 7 0 3 3 4 0 7 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1h k

In the above example all you would need to do is change ‘h’ to the a number to translate the picture left or right OR change ‘k’ to a number to translate

the picture up or down.

ROTATION Matrices:

cos sin 4 4 0 5 5 2 1 3 0 1 4

sin cos 7 4 7 4 7 0 3 3 4 0 7

In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin.

ROTATION Matrices:

cos sin 4 4 0 5 5 2 1 3 0 1 4

sin cos 7 4 7 4 7 0 3 3 4 0 7

In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin.

A

B

Start by creating a picture on a Cartesian coordinate system (preferably

in the first quadrant)

The picture at the right would be represented by the matrix:

Enter this into the Matrix [A] in the

calculator.

Press MATRX , ◄ , ENTER . Change the

dimensions of the matrix to match the points

of your picture.

For our example we will need to change the

dimensions to 2 x 26

To return to the home screen press 2nd

MODE


Matr►list( on the calculator. This function will enable us to put our matrix into the table

of the calculator which can be graphed.

Next, press MATRX , 1 . Before the Matrix can be transformed into the table it has to be turned

vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd

, 1 , , , 2nd

, 2 , ) ,

ENTER .

Next hit 2nd




You can turn the axes on or off by pressing 2nd

, ZOOM and selecting the option AxesOff .

Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar.


Matr►list( on the calculator. This function will enable us to put our matrix into the table

of the calculator which can be graphed.

Next, press, 2 , MATRX , 1 . Before the Matrix can be transformed into the table it has to be

turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd

, 1 , , , 2nd

, 2 ,

) , ENTER .


6 10 10 8 8 6 6 8 8 0 5 8 5 7 7 4 3 5 7 5 5 3 3 3 0 5

10 10 4 4 10 8 3 4 10 0 10 10 6 6 2 1 1 2 2 2 6 4 1 4 0 10

If each screen was viewed in rapid succession it would give the appearance of flying into the picture.

Create your own version of a drawing that you would like to animate using transformational matrices

Using up to 10 frame of animation, show rough sketches of your anticipated slides to create your still animation.

Attempt #1 Attempt #2

Slide # 1 Slide # 2 Slide # 3 Slide # 4

Slide # 5 Slide # 6 Slide # 7 Slide # 8

Slide # 9 Slide # 10

Summary: With a couple of similar pictures we can make an animation of a fish that blows bubbles, an eye that’s winking, or just

about anything you wish to make move. Animations for cartoons, movies, and flip books are created by making several similar

pictures each with a slight change showing where the object has moved too. Creating a fish that moves to the right would require

creating the following pictures:

If these pictures were played in rapid succession on the TI-83 the fish would appear to move to the right. As you create each picture on

the TI-83 you will need to store the picture.

If these pictures were played in rapid succession on the TI-83 the heart would appear to beat. As you create each picture on the TI-83

you will need to store the picture.

Storing your Pictures:

First display the picture you would like to store as your first picture in your animation. You can turn on

a plot by selecting it from the Y= button.

o Press Y = . Highlight Plot1 by pressing the cursor up. Pressing ENTER , will either “turn on”

or “turn off” the plot (if the plot is highlighted then it is turned on). For the “Heart” example

turn Plot1 on and Plot2 off .

o Press GRAPH

With the picture you wish to store on the graph screen, press: 2nd

, PRGM , ◄ , 1 . The

calculator should say StorePic. The TI-82/83/84 allows you to save up to 10 different pictures. Next,

we will need to select where we would like to store the picture. Press: VARS , 4 . Now select

where you would like to store the picture in Pic1 through Pic9. Select Pic1 if this is the first picture.

After selecting the appropriate place to store the picture press ENTER . After pressing enter the

picture that you are storing should re-appear.

DRAW

Next display the picture you would like to store as your second picture in your animation.

o Press Y = . Highlight Plot2 by pressing the cursor up. Pressing ENTER , will either “turn on” or “turn

off” the plot (if the plot is highlighted then it is turned on). For the “Heart” example turn Plot1 off and

Plot2 on .

o Press GRAPH

With the picture you wish to store on the graph screen, press: 2nd

, PRGM , ◄ , 1 . The calculator

should say StorePic. The TI-82/83/84 allows you to save up to 10 different pictures. Next, we will need to

select where we would like to store the picture. Press: VARS , 4 . Now select where you would like to

store the picture in Pic1 through Pic9. After selecting the appropriate place to store the picture press ENTER .

After pressing enter the picture that you are storing should re-appear.

Repeat this process until all of the pictures are stored.

Next, we will need to create a program that displays each of the pictures in rapid succession to give the illusion

of movement. It may help at this point to turn off all of the plots under Y =.

Creating an Animation Program:

Start by creating a new program to do this press: PRGM and select NEW by pressing the ► , ► ,

ENTER and then typing in a name such as “A” , “N” ,“I” ,“M” ENTER .

(FnOff) Now, we should have an almost blank screen ready to be programmed. The first thing we will need to do is force the

calculator to be set up correctly. The first line we will put in a command to turn off any graphed equations. Press VARS , ,

4 ,and 2 ENTER.

(LBL 1) Next, we will need to set up a label so that the program can loop continuously through the

animated sequence. Press: PRGM , 9 , 1 , ENTER.

(ClrDraw) Now, we will need to clear the current screen to begin the animation.

Press: 2nd

, PRGM , 1 , ENTER.

(RecallPic Pic1) Then, we will need to recall the first picture in the animation sequence. Press: 2nd

,

PRGM , ◄ , 2 , VARS , 4 , 1 , ENTER.

(For) If we were to immediately clear this picture and display the next picture the animation would take place too quickly. So,

we will need to set up some type of delay while this picture is being displayed. This can be done with a quick “FOR – Loop” as

shown in example at the right The FOR command is found by pressing PRGM , 4 , X,T,,n , , ,1 , , ,3 ,0 , , ,1 , ) ,

ENTER. The (X, 1, 30, 1) shown in Example 2 stands for (Variable, Beginning Count Number, Ending Count Number, Count

By). PRGM , 7 (End) ENTER . Try changing the Ending Number (30) for different delays.

(ClrDraw) Now, we will need to clear the first picture. Press: 2nd

, PRGM , 1 , ENTER.

(RecallPic Pic2) Then, we will need to recall the first picture in the animation sequence. Press: 2nd

, PRGM , ◄ , 2 ,

VARS , 4 , 2 , ENTER.

Again, if we were to immediately clear this picture & display the next the animation would take place too quickly. We need to

create another “FOR-Loop”. PRGM , 4 , X,T,,n , , ,1 , , ,3 ,0 , , ,1 , ) , ENTER, PRGM , 7 ,ENTER .

If there are additional pictures to include in the animation the procedure is the same for adding more pictures.

(Goto 1) Finally, we need to loop the animation back to the beginning to set up a continuous animation sequence.

Press: PRGM , 0 , 1 . Go back to the HOME SCREEN 2nd

, MODE and Execute PRGM the ANIM program.

Example

DRAW

Press the ON button to interrupt the animation.

PROEJCT – SHEET (INITIAL PICTURES) NAME:

PROEJCT – SHEET (TRANSFORMATIONS) NAME:

Sec 8.5 – Population & Leslie Matrices


Suppose a new species of pig Sus-Gigantis in a small region had the following population

a. Calculate how many 0-2 year olds there will be in the next 5 year cycle.

b. Complete the Table for the next cycle and show your steps.

c. Complete the Table for the next cycle and show your steps.

Age Groups (in years) Year: 2008 0-2 2-4 4-6 6-8 8-10 10-12

# of Pigs in

initial

population

30

25

26

28

22

15

Age Groups

Birth

Rate

Survival

Rate

0-2 0.00 0.60

2-4 0.00 0.85

4-6 1.20 0.90

6-8 0.70 0.80

8-10 0.60 0.50

10-12 0.05 0.00

Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12

in initial pop

(Year:2008)

30

25

26

28

22

15

Population

(Year: 2010)

z Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12

in initial pop

(Year:2010)

Population

(Year: 2012)

DO THIS PROBLEM AGAIN BUT USING MATRICES………….

Suppose a species of pig, Sus-Gigantis, has characteristics as described below:

a. Create the initial population matrix and the corresponding LESLIE matrix.

b. How many Leslie Cycles would have been completed by the year 2010?

c. Using your calculator and the Matrices created in “part a” of this question determine the population of monkeys for

each age group in the year 2010.

d. How would you suggest estimating how many pigs were in each age group in the year 2011?

e. What would be your estimate for the year 2011 using your suggestion from part d?

f. Why is it necessary for the survival rate of the last group to be 0 for this model to work?


Population

(Year: 2010)


Population

(Year: 2011)

Age Groups

Birth

Rate

Survival

Rate

0-2 0.00 0.60

2-4 0.00 0.85

4-6 1.20 0.90

6-8 0.70 0.80

8-10 0.60 0.50

10-12 0.05 0.00

Age Groups (in years) Year: 2008 0-2 2-4 4-6 6-8 8-10 10-12

# of Pigs in

initial

population

30

25

26

28

22

15

2. Suppose a certain species of Elephants has characteristics as described:

a. Calculate how many 0-5 year olds there will be in the next 5 year cycle.

b. Complete the Table for the next cycle and show your steps.

c. Create the initial population matrix and the corresponding LESLIE matrix.

d. How many Leslie Cycles would have been completed by the year 2025?

e. Using your calculator and the Matrices created in “part a” of this question determine the population of elephants for

each age group in the year 2025.

f. It appears that the overall population is declining. If this trend continues how long (or in what year) would you estimate until the

population of all of the subgroup categories finally fall below 1 elephant.

Age Groups (in years) Year: 2000 0-5 5-10 10-15 15-20 20-25 25-30 30-35

# of elephants in

initial

population

20

30

10

30

12

10

6

Age Groups

Birth

Rate

Survival

Rate

0-5 0.00 0.50

5-10 0.10 0.70

10-15 0.40 0.80

15-20 0.90 0.70

20-25 1.10 0.60

25-30 0.30 0.40

30-35 0.00 0

Age Groups (in years) 0-5 5-10 10-15 15-20 20-25 25-30 30-35

in initial pop

(Year:2000)

20

30

10

30

12

10

6

Population

(Year: 2005)


Population

(Year: 2025)

g. BONUS: How would you suggest estimating elephant populations in each age group in 2026?

h. What would be your estimate for the year 2026 using your suggestion from part e?


Population

(Year: 2026)

Sec 8.6 – Markov Chains


A Markov chain is a process that arises naturally in problems that involve a finite number of events or states that change over time.

A company is doing a research study in the metro-Atlanta area about smart phones (either android or

iPhone) and determined that 56% of the sample owned iPhones and the other 44% owned Android based

phones. After following the customers for a while, the researcher determined that each year after that 80%

of the iPhone owners purchased a new iPhone again when it came time to renew their contract. The

remaining 20% switched to Android. The researcher also determined that 65% of android owners purchased

a new Android phone again when it came time to renew their contract. The remaining 35% of android

owners switched to an iPhone.

a) Create a tree diagram showing the percentages of each owner after one cycle of renewals.

b) Create the initial distribution matrix Do

c) Create a Transition Matrix.

d) In the study there were initially 224 iPhone owners and 176 Android owners. If each participant

purchases a new phone every 2 years, determine how many of the participants would own iPhones and

how many would own Androids 8 years after the study began?

Suppose that in the Northwest region of the country, Ford dealers make up 40% of the automobile sales,

GM dealers make up 45% and foreign car dealers 15%. The foreign car dealers decide to offer a no-

interest-for-6-months-after-purchase incentive plan and during the next year of business, the following

transition matrix evolves:

Ford GM Foreign

Ford 0.50 0.20 0.30

GM 0.30 0.40 0.30

Foreign 0.15 0.10 0.75

Based on this matrix, what percentage of GM drivers will switch to Ford each year?

After year 3, what percentage of drivers will be driving Fords?

What percentage of drivers will be driving Fords long-term?

FROM

TO

A food service director for a local high school conducted a survey in hopes of predicting the number of

students who will eat in the cafeteria in the future. The results of the survey are as follows:

If a student eats in the cafeteria on a given day, the probability that he or she will eat there

again the next day is 70% and the probability that he or she will not eat there is 30%.

If a student does not eat in the cafeteria on a given day, the probability that he or she will eat

in the cafeteria the next day is 40% and the probability that he or she will not eat there is 60%.

On Monday, 70% ate at the cafeteria and 30% at somewhere else

a) Create a tree diagram showing the percentages of each owner after one cycle of renewals.

b) Create the initial distribution matrix Do

c) Create a Transition Matrix.

d) Find D3

e) How many students could be expected to eat at

the cafeteria on Friday if there were 350 students

that ate at the cafeteria and 150 that did not on

Monday?

sec 8 - gwinnett county public schools€¦ · sec 8.4 – matrices transformations voting methods...

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