CAMBRIDGE TECHNOLOGY IN MATHS Year 11 TI-89 User guide

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1 Year 11 TI-89 User guide Page 1 of 17 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 for conditions of use Cambridge University Press

2 Year 11 TI-89 User guide Page 2 of 17 A P P E N D I X Computer Algebra System (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. Contents Getting started 1.1 How to calculate using a TI-89 Linear equations and graphs 2.1 How to draw a straight-line graph using a TI How to form a table of values using a TI How to solve simultaneous equations using a TI-89 Statistics 3.1 How to construct a histogram using a TI How to construct a boxplot using a TI How to construct a boxplot with outliers using a TI How to calculate the mean and standard deviation using a TI How to construct a scatterplot using a TI-89 Sequences 4.1 How to use a TI-89 to generate a sequence using recursion Business and related mathematics 5.1 How to investigate simple interest problems using the TI How to investigate compound interest problems using the TI-89

3 Matrices 6.1 How to enter a matrix into a TI How to add, subtract and scalar multiply matrices using a TI How to multiply two matrices using a TI How to find the determinant and inverse of a matrix using a TI 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 Year 11 TI-89 User guide Page 3 of 17 1 Calculation on a TI-89 graphics calculator is done on the Home screen. To call up the Home screen, press HOME. The Home screen consists of an entry line at the bottom, and a history area where processed information is stored. 2 To calculate, simply enter the required expression into the entry line and press Í. For example, if you wish to evaluate , 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) answer 28, not as 3 you might expect. To get the decimal answer, press the diamond key ( ) before pressing Í. Entry line History area 4 To clear the history area, press & (Tools), use the down arrow key D to move to 8 :Clear Home, and press Í. The history area of the screen is now cleared.

4 Year 11 TI-89 User guide Page 4 of 17 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. 1 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. 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 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 0 enables 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).

5 Year 11 TI-89 User guide Page 5 of 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. 1 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=1and 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 1 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.

6 Year 11 TI-89 User guide Page 6 of 17 2 a Complete the command by typing in: y=2x+6andy=--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 = 3 4 and y = 9 2. 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 = 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: Enter the data into your calculator using the Stats/List Editor. a Press Œ,movetoStats/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) tomovethe cursor to highlight list1 and press. Use the down arrow key (D)tomovethe 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.

7 c With Plot 1 highlighted, press & (Define). This will take you to the Define Plot 1 dialogue box. d Complete the dialogue box as follows: For Plot Type, select Histogram (B ). For x, type in list1 (y7list ƒ1). 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 is 25 4 = 3.5, a convenient width. 6 3 Pressing Í confirms your selection and returns you to the Plot Setup menu. Year 11 TI-89 User guide Page 7 of 17 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.

8 Year 11 TI-89 User guide Page 8 of 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) Use a TI-89 graphics calculator to construct a boxplot. 1 Enter the data into your calculator using the Stats/List Editor. a Press Œ,movetoStats/List Editor and 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) tomovethe cursor to highlight list1 and press. Use the down arrow key (D)tomovethe cursor back down into the data column and enter the data. 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. c Press & to Define Plot 1. d Complete the dialogue box as follows: For Plot Type, select Box Plot (B Â). For x, type in list1. 4 Pressing Í confirms your selection and returns you to the Plot Setup menu. 5 Pressing (ZoomData)inPlot Setup automatically plots the boxplot in a properly scaled window.

9 Year 11 TI-89 User guide Page 9 of 17 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: Enter the data into your calculator using the Stats/List Editor. a Press Œ,movetoStats/List Editor and press Í to select. b Type the data into list1. 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. c Press & to Define Plot 1. d Complete the dialogue box as follows: For Plot Type, select Mod Box Plot (B ). Leave Mark as Box. For x, type in list1. 3 Pressing Í confirms your selection and returns you to the Plot Setup menu.

10 Year 11 TI-89 User guide Page 10 of 17 5 Pressing (ZoomData)inPlot Setup automatically 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) 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. 1 Enter the data into your calculator using the Stats/List Editor. a Press Œ,movetoStats/List Editor and press Í to select. b Type the data into list1. Your screen should 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.

11 Year 11 TI-89 User guide Page 11 of 17 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 = Q 3 Q 1 = = 9.5 Range = Max Min = = 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 Mark Use a graphics calculator to construct a scatterplot. Treat Time as the independent (x-) variable. 1 Enter the data into your calculator using the Stats/List Editor. a Press Œ,movetoStats/List Editor and 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. 3 Set up the calculator to plot a statistical graph. a Press to access the Plots menu. b Press Í to select 1:Plot Setup. c Press & to Define Plot 1. d Complete the dialogue box as follows: For Plot Type, select Scatter (B À). Leave Mark as Box. For x, type in list1.fory, type in list2. IV: list1 (x ) DV: list2 (y )

12 Year 11 TI-89 User guide Page 12 of 17 4 Pressing Í confirms your selection and returns you to the Plot Setup menu. 5 Pressing (Zoom Data)inPlot Setup automatically plots the seatterplot in a properly scaled window. 4.1 How to use a TI-89 to generate a sequence using recursion a b Use recursion to generate the first five terms of the following arithmetic sequence on a TI-89 graphics calculator: 2, 9, 16,... Find the value of the 10th term. 1 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 Pressing Í again generates the next term, 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.

13 Year 11 TI-89 User guide Page 13 of How to investigate simple interest problems using the TI-89 How much interest is earned if $ is invested at 8.25% simple for 10 years? 1 Substitute P = and r = 8.25 into the formula for simple interest. 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. I = Prt 100 = t 100 = 825t Write your answer. Interest earned = $ 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. 1 Substitute P = $ and r = 8 into the formula for compound interest. ) t ( A = P 1 + r 100 ( = ) t

14 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) x and press Í. 3 Press 0 to view a table of values. Year 11 TI-89 User guide Page 14 of 17 a By scrolling down the table, we can readily see that after 3 years an amount of $2519 will have accumulated. Write your answer. b To determine the amount of interest earned, subtract the amount invested. Write 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. a Amount of investment = $2519 b Interest earned = A P = $ = $ How to enter a matrix into a TI-89 [ ] Enter the matrix A = into a graphics calculator. 1 Press HOME to take you to the Home screen.

15 2 On the Home screen, matrices are entered row by row. [ ] To enter the matrix A = : type in [2, 3, 0; 1, 4, 2], press X then type a press 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 Year 11 TI-89 User guide Page 15 of How to add, subtract and scalar multiply matrices using a TI-89 [ ] If A = [ ] and B =, find: a A + B b A B c 3A 2B 1 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+band press Í to evaluate. [ ] Write your answer. A + B = b To calculate A B, type in a-band press Í to evaluate. [ ] Write your answer A B = 1 6 1

16 c To calculate 3A 2B, type in 3a-2band press Í to evaluate. Year 11 TI-89 User guide Page 16 of 17 [ ] Write your answer A 2B = How to multiply two matrices using a TI-89 [ ] [ ] If C = and D =, find the matrix product CD 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. CD = [ ] How to find the determinant and inverse of a matrix using a TI If A = 4 1 0, find det(a) and A 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.

17 Year 11 TI-89 User guide Page 17 of 17 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 2 = How to use the TI-89 to solve a set of two simultaneous linear equations using the inverse matrix Solve the following set of simultaneous linear equations using matrix methods: 3x + 4y = 6 2x + 3y = 4 1 Rewrite the equations in matrix form. 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 1 C. [ ][ ] [ ] 3 4 x 6 = 2 3 y 4 [ ] [ ] [ ] 3 4 x 6 Let A =, X =, C =. 2 3 y 4 Then A X = C or X = A 1 C (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 1 C. Type in a -1 c and store in x (ƒñzìàƒó ƒ [x]). Press Í to evaluate. Write your answer. X = [ ] 2 or x = 2andy= 0. 0