cambridge technology in maths...cambridge technology in maths year 11 ti-89 user guide contents...

CAMBRIDGE TECHNOLOGY IN MATHS

Year 11 TI-89 User guide

CONTENTS Getting started 2 Linear equations and graphs 3 Statistics 5 Sequences 11 Business and related mathematics 12 Matrices 13

Published in: Cambridge Essential Standard General Mathematics © Peter Jones, Kay Lipson, David Main, Barbara Tulloch 2005

See www.technologyinmaths.com.au for conditions of use

Cambridge University Press

Year 11 TI-89 User guide of 17

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0521672600Xapx.xml CUAT016-EVANS November 4, 2005 6:44

A P P E N D I X

Computer AlgebraSystem (TI-89 Graphics

Calculators)

This supplement has been specifically written to support the use of the TI-89 family of

CAS-based calculators (which is also taken to include TI-92 and the Voyage 200) in General

Mathematics. The screens have been taken from a TI-89 Titanium.

ContentsGetting started

1.1 How to calculate using a TI-89

Linear equations and graphs2.1 How to draw a straight-line graph using a TI-89

2.2 How to form a table of values using a TI-89

2.3 How to solve simultaneous equations using a TI-89

Statistics3.1 How to construct a histogram using a TI-89

3.2 How to construct a boxplot using a TI-89

3.3 How to construct a boxplot with outliers using a TI-89

3.4 How to calculate the mean and standard deviation using a TI-89

3.5 How to construct a scatterplot using a TI-89

Sequences4.1 How to use a TI-89 to generate a sequence using recursion

Business and related mathematics5.1 How to investigate simple interest problems using the TI-89

5.2 How to investigate compound interest problems using the TI-89

465


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466 Essential Standard General Mathematics

Matrices6.1 How to enter a matrix into a TI-89

6.2 How to add, subtract and scalar multiply matrices using a TI-89

6.3 How to multiply two matrices using a TI-89

6.4 How to find the determinant and inverse of a matrix using a TI-89

6.5 How use the TI-89 to solve a set of two simultaneous linear equations using the inverse

matrix

1.1 How to calculate using a TI-89

1 Calculation on a TI-89 graphics calculator is

done on the Home screen. To call up the Homescreen, press HOME .

The Home screen consists of an entry line at

the bottom, and a history area where

processed information is stored.

History area

Entry line

2 To calculate, simply enter the required

expression into the entry line and press Í.

For example, if you wish to evaluate

5 + 3 × 6, type the expression into the entry

line and press Í. You can move between

the entry line and the history area using the

arrow keys (A, C, B, D).

3 The TI-89 is set to give exact answers to a

computation rather than its decimal

approximation. For example, if you ask the

calculator to evaluate 56 ÷ 6, you will get the

exact (fractional) answer28

3, not 9.333. . . as

you might expect.

To get the decimal answer, press the diamond

key (¥) before pressing Í.

4 To clear the history area, press & (Tools), use

the down arrow key D to move to 8 :ClearHome, and press Í. The history area of

the screen is now cleared.


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Appendix — Computer Algebra Systems (TI-89 Graphics Calculators) 467

5 To clear the entry line, press

:.

2.1 How to draw a straight-line graph using a TI-89

Use a TI-89 graphics calculator to draw the graph of y = 8 − 2x .

Steps1 Enter the equation using the Y= screen.

Press ¥# (above the & key) to open

the Y= screen.

With the cursor opposite y1, type in 8 - 2xand press Í.

Note: When you start typing, the cursor jumps to the

entry line at the bottom of the screen for you to type

in the expression.

2 Press „ (ZOOM) ¸ to obtain the graph.Note: „ (ZOOM) ¸ automatically plots the graph in

the standard viewing window [−10, 10] by

[−10, 10]; that is, −10 ≤ x ≤ 10 and −10 ≤ y ≤ 10.

3 To get a graph more like the one plotted by

hand, press ¥$ and change the

settings as shown.Note 1: Making xmin and ymin = -0.5, rather than 0enables you to see the axes. Use the Ì key here.

Note 2: xscl and yscl set the distances between the

tick marks on the x- and y axes.

4 Press ¥% (above the ( key) to obtain

the graph.

5 Press ( (Trace) to place the cursor on

the graph. The coordinates of the point

(2.28, 3.43) are shown.Note: You can move the cursor along the graph using

the arrow keys (Band A).


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2.2 How to form a table of values using a TI-89

Use a TI-89 graphics calculator to form a table of values for the relationship y = 8 − 2x .

Steps1 Enter the equation using the Y= screen.

� Press ¥# (above the & key) to open

the Y= screen.� With the cursor opposite y1, type in 8 - 2x

and press Í.

2 Press ¥ 0 to obtain the table.Note: You can move the cursor up, down and across

the table using the arrow keys (B, D, C and A).

3 Normally, the table is set to start at x = 1 and

increases in steps of 1. If this is not the case,

press ¥ - (above the † key) to open

the TBLSET (table set-up) screen and adjust it

so that tblStart = 1 and Δtbl = 1. Press ¥

0 to return to the table.

2.3 How to solve simultaneous equations using a TI-89

Use a TI-89 graphics calculator to solve the following pairs of simultaneous equations.

a y = 2x + 6

y = −2x + 3

b x + y = 3

2x − y = 9

Steps1 Simultaneous equations can be solved using

solve( in the Algebra menu.� Press HOME to take you to the Home

screen. Clear if necessary.� Press „ (Algebra) to open the Algebra

menu and press Í to select 1:solve(.This pastes the solve( command into the

entry line.


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2 a Complete the command by typing in:

y = 2x + 6 and y = --2x + 3, x)Note: To obtain the ‘and’ with space either side,

press ƒO, type in and then press ƒO.

Pressing Í to solve gives the solutions

as x = − 34

and y = 92.

Pressing ¥Í to solve gives the

solutions as x = −0.75 and y = 4.5.

b Complete the command by typing in:

x + y = 3 and 2x - y = 9, x)Pressing Í to solve gives the solutions

as x = 4 and y = −1.

3.1 How to construct a histogram using a TI-89

Use a TI-89 graphics calculator to display the following set of marks in the form of a

histogram:

16 11 4 25 15 7 14 13 14 12 15 13 16 14

15 12 18 22 17 18 23 15 13 17 18 22 23

Steps1 Enter the data into your calculator using the

Stats/List Editor.a Press Œ, move to Stats/List Editor and

press Í to select. If you have a

standard TI-89, you may have to download

this application from the web.

b Type the data into list1. If there is data in

list1, use the up arrow key (C) to move the

cursor to highlight list1 and press ‘.

Use the down arrow key (D) to move the

cursor back down into the data column and

enter the data. Your screen should look like

the one shown.

2 Set up the calculator to plot a statistical graph.

a Press „ to access the Plots menu.

b Press Í to select 1: Plot Setup. This

will take you to the Plot Setup menu.


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c With Plot 1 highlighted, press & (Define).

This will take you to the Define Plot 1dialogue box.

d Complete the dialogue box as follows:

For Plot Type, select Histogram (B¶).

For x, type in list1 (y7 listƒ1).

For His.Bucket Width, type in 3.5.Note: ‘Bucket width’ means ‘interval width’. Choose

a minimum of five intervals and divide the range of

values by the number of intervals to get the ‘bucket

width’. If we choose six intervals, the bucket width is25 − 4

6= 3.5, a convenient width.

3 Pressing Í confirms your selection and

returns you to the Plot Setup menu.

4 Set the viewing window (¥$) with

the following entries.xmin = 4

xmax = 29

xscl = 0 (no tick marks will appear on the scale)

ymin = −5 (to allow space below the histogram)

ymax = 13 (a first guess at the maximum height of

the histogram; half the number of data values)

yscl = 1 (leave as is)

xres = 2 (leave as is).

5 Pressing ¥% plots the histogram.Note: Before graphing the histogram, you should

make sure that all graphs in the Y= screen have been

switched off or cleared.

6 Pressing ( (Trace) places a marker at the top

of the first column of the histogram and tells

us that the first class interval contains all

values ranging from 4 to less than 7.5. For this

interval, the count is two (n:2).

To find out the counts in the other intervals,

use the horizontal arrow key (B) to hop from

interval to interval.


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3.2 How to construct a boxplot using a TI-89

The following are the monthly rainfall figures for a year in Melbourne:

Month J F M A M J J A S O N D

Rainfall (mm) 48 57 52 57 58 49 49 50 59 67 60 59

Use a TI-89 graphics calculator to construct a boxplot.



press Í to select.

b Type the data into list1. Your screen should

look like the one shown. If there is data in

list1, use the up arrow key (C) to move the

cursor to highlight list1 and press ‘.

Use the down arrow key (D) to move the

cursor back down into the data column and

enter the data.



b Press Í to select 1:Plot Setup.

c Press & to Define Plot 1.


For Plot Type, select Box Plot (BÂ).

For x, type in list1.



5 Pressing ‡ (ZoomData) in Plot Setupautomatically plots the boxplot in a properly

scaled window.


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6 Key values can be read from the boxplot by

pressing ( (Trace). This places a marker on

the boxplot. You can then use the horizontal

arrow keys (A and B) to move from point to

point on the boxplot and read off the

associated values.

Starting at the far left of the plot, we see that

the:

minimum value is 48 (min X:48)first quartile is 49.5 (q1:49.5)median is 57 (Med:57)third quartile is 59 (q3:59)maximum value is 67 (max X:67).

3.3 How to construct a boxplot with outliers using a TI-89

Use a TI-89 graphics calculator to display the following set of marks in the form of a

boxplot with outliers:

28 21 21 3 22 31 35 26 27 33

36 35 23 24 43 31 30 34 48



press Í to select.


look like the one shown.






For Plot Type, select Mod Box Plot (B

·).

Leave Mark as Box.

For x, type in list1.




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5 Pressing ‡ (ZoomData) in Plot Setupautomatically plots the boxplot in a properly

scaled window.

6 Key values can be read from the boxplot by

pressing ( (Trace). This places a marker on

the boxplot. You can then use the horizontal

arrow keys (A and B) to move from point to

point on the boxplot and read off the

associated values

Starting at the far left of the plot, we see that

the:

minimum value is 3 (min X:3) – it is also

an outlier.

lower adjacent value is 21 (X:21)

first quartile is 23 (q1:23)

median is 30 (Med:30)

third quartile is 35 (q3:35)

maximum value is 48 (max X:48).

3.4 How to calculate the mean and standard deviation using a TI-89

The following are the monthly rainfall figures for a year in Melbourne:

Month J F M A M J J A S O N D

Rainfall (mm) 48 57 52 57 58 49 49 50 59 67 60 59

Use a TI-89 graphics calculator to determine the mean and standard deviation, median

and interquartile range, and range for this data set. Give your answers correct to

1 decimal place where necessary.



press Í to select.


look like the one shown.

2 Calculate the mean and standard deviation.

a Press F4 to access the Calc(ulate) menu.

b With 1:1-Var Stats highlighted, press

Í to select. This will take you to the

1-Var Stats dialogue box.


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3 Complete the dialogue box:

For List:, type in list1.Note: This is not necessary if list1 is already

shown.

4 Press Í and use † to view the results.Note: The value of the standard deviation is

given by Sx.

Write your answers to the required

degree of accuracy (2 decimal places).

x = 55.4; s = 5.8 (correct to 1 d.p.)

M = 57; IQR = Q3 − Q1 = 59 − 49.5 = 9.5

Range = Max − Min = 67 − 48 = 19

3.5 How to construct a scatterplot using a TI-89

The data below gives the marks that students obtained on an examination and the times

they spent studying for it.

Time 4 36 23 19 1 11 18 13 18 8

Mark 41 87 67 62 23 52 61 43 65 52

Use a graphics calculator to construct a scatterplot. Treat Time as the independent (x-)

variable.



press Í to select.

b Type the data into list1 and list2. Your

screen should look like the one shown.

2 Identify and write down the independent (IV)

and dependent (DV) variables. Label as x and

y respectively.Note: You need this information for step 3.

IV : list1 (x )

DV : list2 (y )






For Plot Type, select Scatter (BÀ).

Leave Mark as Box.

For x, type in list1. For y, type in list2.


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5 Pressing ‡ (Zoom Data) in Plot Setupautomatically plots the seatterplot in a

properly scaled window.

4.1 How to use a TI-89 to generate a sequence using recursion

a Use recursion to generate the first five terms of the following arithmetic sequence

on a TI-89 graphics calculator: 2, 9, 16, . . .

b Find the value of the 10th term.

Steps1 Write the value of first term. a = 2

2 Determine the common difference. d = 9 − 2 = 7

3 Press HOME to take you to the Home screen.

Type in 2, the value of the first term. Press

Í. The calculator stores the value 2 as

Ans(wer) (you can’t see this yet).

4 Now type in +7. Press Í. The second

term in the sequence, 9, is generated and

stored as Ans(wer).Note: ans(1) was automatically inserted before +7.

5 Pressing Í again generates the next term,

16.

6 a Keep pressing Í until five terms are

generated.

Write your answer. a The first five terms are : 2, 9, 16, 23, 30.

b Keep pressing Í until the

10th term is generated.

Write your answer. b The 10th term is 65.


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5.1 How to investigate simple interest problems using the TI-89

How much interest is earned if $10 000 is invested at 8.25% simple for 10 years?

Steps1 Substitute P = 10 000 and r = 8.25 into the

formula for simple interest.

I = P r t

100= 10 000 × 8.25 × t

100

= 825t

2 Because the calculator expects formulas to be

defined in terms of x and y, we will enter this

rule as shown.

To enter the rule:

press ¥# to open the Y= menu.

opposite y1 type in 825x and press Í.

3 Press ¥0 to view a table of values.

By scrolling down the table, we can readily

see that after 10 years interest of $8250 will

have been earned.

Write your answer. Interest earned = $8250

4 A graph of the simple interest function can

also be constructed using the graphics

calculator.

a Press ¥$ and set the viewing

window as shown.

b Press ¥% to graph.

c Press … (Trace) to place a marker on the

graph and use the arrow keys (B, A) to

move along the graph to find an

approximate solution. When in Trace mode

an exact solution can be found by typing in

the x value then pressing Í.Note: The interest rate is equal to the slope of this line.

5.2 How to investigate compound interest problems using the TI-89

a Determine the amount of money accumulated after 3 years if $2000 is invested at an

interest rate of 8% per annum, compounded annually. Give your answer to the

nearest dollar.

b Determine the amount of interest earned.Steps1 Substitute P = $10 000 and r = 8 into the

formula for compound interest.

A = P

(1 + r

100

)t

= 2000 ×(

1 + 8

100

)t


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2 Because the calculator expects formulas to be

defined in terms of x and y, we will enter this

rule as shown.

To enter the rule:

press ¥# to open the Y= menu.

opposite Y1 type 2000∗ (1 + 8/100)∧xand press Í.

3 Press ¥ 0 to view a table of values.

a By scrolling down the table, we can readily

see that after 3 years an amount of $2519

will have accumulated.

a Amount of investment = $2519Write your answer.

b To determine the amount of interest earned,

subtract the amount invested.b Interest earned = A − P

= $2519 − 2000

= $519Write your answers.

4 A graph of the compound interest function

can also be constructed using the graphics

calculator.

a Press ¥$ and set the viewing

window as shown.

b Press ¥% to graph.

c Press ( (Trace) to place a marker on the

graph and use the arrow keys (B, A) to

move along the graph to find an

approximate solution. A more exact

solution can be found by pressing ‡

(Math) À (Value) and typing in 3 Í.

6.1 How to enter a matrix into a TI-89

Enter the matrix A =[

2 3 0

1 4 2

]into a graphics calculator.

Steps1 Press HOME to take you to the Home screen.


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2 On the Home screen, matrices are entered row

by row.

To enter the matrix A =[

2 3 0

1 4 2

]:

type in [2, 3, 0; 1, 4, 2], press X then

type apress Í to set up the matrix and

display it on the Home screen.

Your screen should now look like that shown

opposite. Until you redefine a, whenever you

type a into the entry line, it will represent the

matrix

[2 3 0

1 4 2

].

6.2 How to add, subtract and scalar multiply matrices using a TI-89

If A =[

2 3 0

1 4 2

]and B =

[1 0 3

2 −2 1

], find:

a A + B b A − B c 3A − 2B

Steps1 Enter the matrices A and B into your

calculator.Note: If you have been following the text, these

matrices may already be stored in your calculator.

Refer to 6.1 if you are unsure how to enter a matrix

into your calculator.

2 a To calculate A + B, type in a + b and press

Í to evaluate.

Write your answer. ∴ A + B =[

3 3 3

3 2 3

]

b To calculate A − B, type in a - b and press

Í to evaluate.

Write your answer. ∴ A − B =[

1 3 −3

−1 6 1

]


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c To calculate 3A − 2B, type in 3a - 2b and

press Í to evaluate.

Write your answer. ∴ 3A − 2B =[

4 9 −6

−1 16 4

]

6.3 How to multiply two matrices using a TI-89

If C =[

11 5

10 9

]and D =

[6

1

], find the matrix product CD.

Steps1 Enter the matrices C and D into your

calculator.Note: Refer to 6.1 if you are unsure how to do this.

2 To calculate CD, type in c ∗ d and press

Í to evaluate.Note: When multiplying matrices, it is necessary to

include the multiplication sign.

Write your answer. ∴ C D =[

71

69

]

6.4 How to find the determinant and inverse of a matrix using a TI-89

If A =

⎡⎢⎣1 2 3

4 1 0

2 0 2

⎤⎥⎦, find det(A) and A−1.

Steps1 Enter the matrix A into your calculator.

Note: Refer to 6.1 if you are unsure how to do this.

2 To calculate det(A), type in det(a) and press

Í to evaluate.


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Write your answer. ∴ det(A ) = −20

3 To calculate the inverse matrix A−1, type in

a∧-1 (ƒÑZÌÀ) and press Í to

evaluate.

Write your answer.

∴ A −1 =

⎡⎢⎢⎢⎢⎢⎢⎣

− 1

10

1

5

3

202

5

1

5−3

51

10− 1

5

7

20

⎤⎥⎥⎥⎥⎥⎥⎦

6.5 How to use the TI-89 to solve a set of two simultaneous linear equations using theinverse matrix

Solve the following set of simultaneous linear equations using matrix methods:

3x + 4y = 6

2x + 3y = 4

Steps1 Rewrite the equations in matrix

form.

[3 4

2 3

] [x

y

]=

[6

4

]

2 By identifying the matrices A, X, and C,

rewrite the matrix equation in the form

AX = C . Provided det(A) �= 0, the solution is

X = A−1C .

Let A =[

3 4

2 3

], X =

[x

y

], C =

[6

4

].

Then A X = C

or X = A −1C (det(A ) �= 0)

3 Enter the matrices A and C into your

calculator.Note: Refer to 6.1 if you are unsure how to do this.

4 Solve the matrix equation by evaluating the

matrix product A−1C.

Type in a∧-1 × c and store in x(ƒÑZÌÀƒÓ¿ƒ [x]).

Press Í to evaluate.

Write your answer. ∴ X =[

2

0

]or x = 2 and y = 0.


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