Part IV: Texas Instruments TI-89 IV.1 Getting started with the TI-89

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In this guide, the key with the green diamond symbol inside a green border will be indicated by key with the white arrow pointing up inside a white border (the shift key) will be indicated by the key with the white arrow (the backspace key) pointing to the left will be indicated by.
There are 5 blue keys below the calculator screen labeled F1 through F5. These function keys have different effects depending on the screen that is currently showing. The effect or menu of the function keys corresponding to a screen are shown across the top of the display. IV.1.1 Basics: Press the ON key to begin using your TI-89. If you need to adjust the display contrast, first press and hold , then press (the minus key) to lighten or + (the plus key) to darken. When you have finished with the calculator, turn it off to conserve battery power by pressing 2nd and then OFF. Check our TI-89s settings by pressing MODE. If necessary, use the arrow keys to move the blinking cursor to a setting you want to change. You can also use F1 to go to page 1, F2 to go to page 2, and F3 to go to page 3 of the MODE menu. To change a setting, use to get to the setting that you want to change, then press to see the options available. Use or to highlight the setting you want and press ENTER to select the setting. To start with, select the options shown in Figures IV.1, IV.2, and IV.3: function graphs, main folder, floating decimals with 10 digits displayed, radian measure, normal exponential format, real numbers, rectangular vectors, pretty print, full screen display, home screen showing, approximate calculation mode, decimal base, international system of units (metric measurement), and English. Note that some of the lines on page 2 and page 3 of the MODE menu are not readable. These lines pertain to options that are not set as above. Details on alternative options will be given later in this guide. For now, leave the MODE menu by pressing HOME or 2nd QUIT. Some of the current settings are shown on the status line of the home screen.
Figure IV.1: MODE menu, page 1
Figure IV.2: MODE menu, page 2
Figure IV.3: MODE menu, page 3
Graphing Technology Guide
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IV.1.2: Editing: One advantage of the TI-89 is that you can use the arrow keys to scroll in order to see a long calculation. For example, type this sum (Figure IV.4): Then press ENTER to see the answer. The sum is too long for both the entry line and the history area. The direction(s) in which the line extends off the screen is indicated by an ellipsis at the end of the entry line and arrows or in the history area. You can scroll through the entire calculation by using or to put the cursor on the appropriate line and then using or to move the cursor to the part of the calculation that you wish to see.
Figure IV.4: Home screen Often we do not notice a mistake until we see how unreasonable an answer is. The TI-89 permits you to redisplay an entire calculation, edit it easily, then execute the corrected calculation. Suppose you had typed 56 as in Figure IV.5 but had not yet pressed ENTER, when you realize that 34 should have been 74. Simply press as many times as necessary to move the blinking cursor line until it is to the immediate right of the 3, press to delete the 3, and then type 7. On the other hand if 34 should have been 384, move the cursor until it is between the 3 and the 4 and then type 8. If the 34 should have been 3 only, move the cursor to the right of the 4, and press to delete the 4.
Figure IV.5: Editing a calculation Technology Tip: The TI-89 has two different inputting modes: insert and overtype. The default mode is insert mode, in which the cursor is a blinking vertical line and new text will be inserted at the cursors position and other characters are pushed to the right. In the overtype mode, the cursor is a blinking square and the characters that you type replace the existing characters. To change from one mode to another, press 2nd INS. The TI-89 remains in whatever the last input mode was, even after being turned off.
TI-89 Graphics Calculator
Even if you had pressed ENTER, you may still edit the previous expression. Immediately after you press ENTER your entry remains on the entry line. Pressing moves the cursor to the beginning of the line, while pressing puts the cursor at the end of the line. Now the expression can be edited as above. To edit a previous expression that is no longer on the entry line, press 2nd and then ENTRY to recall the prior expression. Now you can change it. In fact, the TI-89 retains as many entries as the current history area holds in a last entry storage area, including entries that have scrolled off the screen. Press 2nd ENTRY repeatedly until the previous line you want is on the entry line. (The number of entries that the history area can hold may be changed, see your users manual for more information.) To clear the entry line, press CLEAR while the cursor is on that line. To clear previous entry/answer pairs from the history area, use or to move the cursor to either the entry or the answer and press CLEAR (both the entry and the answer will be deleted from the display). To clear the entire history area, press F1 [Tools] 8 [Clear Home], although this will not clear the entry line. Technology Tip: When you need to evaluate a formula for different values of a variable, use the editing feature to simplify the process. For example, suppose you want to find the balance in an investment account if there is now $5000 in the account and interest is compounded annually at the rate of 8.5%. The formula r nt for the balance is P 1 , where P principal, r rate of interest (expressed as a decimal), n n number of times interest is compounded each year, and t number of years. In our example, this becomes 50001 .085t. Here are the keystrokes for finding the balance after t 3, 5, and 10 years (results are shown in Figure IV.6).

Years 10

Keystrokes 5000 (1 +.085) ^ 3 ENTER 5 ENTER 10 ENTER
Balance $6386.45 $7518.28 $11,304.92
Figure IV.6: Editing expressions Then to find the balance from the same initial investment but after 5 years when the annual interest rate is 7.5%, press the following keys to change the last calculation above: ENTER. You could also use the CLEAR key to erase everything to the right of the current location of the cursor. Then, changing the calculation from 10 years at the annual interest rate of 8.5% to 5 years at the annual interest rate of 7.5% is then done by pressing CLEAR ENTER. IV.1.3 Key Functions: Most keys on the TI-89 offer access to more than one function, just as the keys on a computer keyboard can produce more than one letter (g and G) or even quite different characters (5 and %). The primary function of a key is indicated on the key itself, and you access that function by a simple press on the key. To access the second function indicated in yellow or to the left above a key, first press 2nd (2nd appears on the status line) and then press the key. For example, to calculate 25 press 2nd 25 ) ENTER.
Technology Tip: The TI-89 automatically places a left parenthesis, (, after many functions and operators (including 2nd LN, e x , 2nd SIN, 2nd COS, 2nd TAN, and 2nd ). If a right parenthesis is not entered, the TI-89 will respond with an error message indicating that the right parenthesis is missing. When you want to use a function printed in green or to the right above a key, first press ( appears on the status line) and then press the key. For example, if you are in exact calculation mode and want to find the approximate value of 45 press 2nd 45 ). The TI-89 can produce both upper and lower case letters. When you want to use a lower case letter printed in purple or to the right above a key, first press alpha (a lower case a appears on the status line) and press the key. For example, to use the letter k in a formula, press alpha K. If you need several letters in a row, press 2nd a-lock, and then press all the letters you want. Remember to press alpha when you are finished and want to restore the keys to their primary functions. To type upper case letters, press and then press the letter. To lock in upper case letters, press alpha (an upper case A appears on the status line). To restore the keys to their primary functions, press alpha. Technology Tip: There are separate keys for the commonly used letters X, Y, Z, and T. A simple press of the key will produce a lower case letter while pressing and then the key will produce an upper case letter. IV.1.4 Order of Operations: The TI-89 performs calculations according to the standard algebraic rules. Working outwards from inner parentheses, calculations are performed from left to right. Powers and roots are evaluated first, followed by multiplications and divisions, and then additions and subtractions. Note that the TI-89 distinguishes between subtraction and the negative sign. If you wish to enter a negative number, it is necessary to use the (-) key. For example, you would calculate 3 by pressing (-) 5 (4 (-) 3) ENTER to get 7. Enter these expressions to practice using your TI-89. Expression Keystrokes 3 ENTER Display --36

Figure IV.12: Y= screen

Figure IV.13: Evaluating a function
Assign the value 2230 to the variable x by using these keystrokes (see Figure IV.13): 2230 STO X ENTER. Then press the following keystrokes to evaluate y1 and find Januarys wages: Y 1 ( X ) ENTER, completes the calculation. It is not necessary to repeat all these steps to find the February wages. Simply press to begin editing the previous entry, change X to 1865, and press ENTER (see Figure IV.13). You may also have TI-89 make a table of values for the function. Press TblSet to set up the table (Figure IV.14). Move the blinking cursor down to the fourth line beside Independent, then press and 2 [ASK] ENTER. This configuration permits you to input values for x one at a time. Now press TABLE or APPS 5 [Table], enter 2230 in the x column, and press ENTER (see Figure IV.15). Press to move to the next line and continue to enter additional values for x. The TI-89 automatically completes the table with the corresponding values of y1. Press 2nd QUIT to leave the TABLE screen.
Figure IV.14: TABLE SETUP screen
Figure IV.15: Table of values
Technology Tip: The TI-89 requires multiplication to be expressed between variables, so xxx does not mean x3, rather it is a new variable named xxx. So, you must use either s between the xs or ^ for powers of x. Of course, expressed multiplication is not required between a constant and a variable. See your TI-89 manual for more information about the allowed usage of implied multiplication. IV.2.2 Functions in a Graph Window: Once you have entered a function in the Y= screen of the TI-89, just press GRAPH to see its graph. The ability to draw a graph contributes substantially to our ability to solve problems. For example, here is how to graph y x3 4x. First press Y= and delete anything that may be there by moving with the arrow keys to y1 or to any of the other lines and pressing CLEAR wherever necessary. Then, with the cursor on the (now cleared) top line (y1), press (-) X ^ 3 + 4 X ENTER to enter the function (as in Figure IV.16). Now press GRAPH and the TI-89 changes to a window with the graph of y x3 4x (Figure IV.18). While the TI-89 is calculating coordinates for a plot, it displays the word BUSY on the status line.
Technology Tip: If you would like to see a function in the Y= menu and its graph in a graph window, both at the same time, press MODE to open the MODE menu and press F2 to go to the second page. The cursor will be next to Split Screen. Select either TOP-BOTTOM or LEFT-RIGHT by pressing and 2 or 3, respectively. Now the 2 lines below the Split 1 App line have become readable, because these options apply only when the calculator is in the split screen mode. The Split 1 App will automatically be the screen you were on prior to pressing MODE. You can choose what you want the top or left-hand screen to show by moving down to the Split 1 App line, pressing and the number of the application you want in that window. The Split 2 App determines what is shown in the bottom or right-hand window. Press ENTER to confirm your choices and your TI-89s screen will now be divided either horizontally or vertically (as you choose). Figure IV.16 show the graph and the Y= screen with the settings shown in Figure IV.17. The split screen is also useful when you need to do some calculations as you trace along a graph. In split screen mode, one side of the screen will be more heavily outlined. This is the active screen, i.e., the screen that you can currently modify. You can change which side is active by using 2nd to access the symbol above the APPS key. For now, restore the TI-89 to Full screen. Technology Tip: Note that if you set one part of your screen to contain a table and the other to contain a TblSet to generate a new graph, the table will not necessarily correspond to the graph unless you use table based on the functions being graphed (as in Section IV.2.1).

Figure IV.16: Split screen: LEFT-RIGHT
Figure IV.17: MODE settings for Figure IV.16
Your graph window may look like the one in Figure IV.18 or it may be different. Because the graph of y x3 4x extends infinitely far left and right and also infinitely far up and down, the TI-89 can display only a piece of the actual graph. This displayed rectangular part is called a viewing window. You can easily change the viewing window to enhance your investigation of a graph.
Figure IV.18: Graph of y x3 4x The viewing window in Figure IV.18 shows the part of the graph that extends horizontally from 10 to 10 and vertically from 10 to 10. Press WINDOW to see information about your viewing window. Figure IV.19 shows the WINDOW screen that corresponds to the viewing window in Figure IV.18. This is the standard viewing window for the TI-89.
The variables xmin and xmax are the minimum and maximum x-values of the viewing window; ymin and ymax are the minimum and maximum y-values. xscl and yscl set the spacing between the tick marks on the axes. xres sets pixel resolution (1 through 10) for function graphs.
Figure IV.19: Standard WINDOW Technology Tip: Small xres values improve graph resolution, but may cause the TI-89 to draw graphs more slowly. Use and to move up and down from one line to another in this list; pressing the ENTER key will move down the list. Press CLEAR to delete the current value and then enter a new value. You may also edit the entry as you would edit an expression. Remember that a minimum must be less than the corresponding maximum or the TI-89 will issue an error message. Also, remember to use the (-) key, not (which is subtraction), when you want to enter a negative value. Figures IV.18-19, IV.20-21, and IV.22-23 show different WINDOW screens and the corresponding viewing window for each one.
Figure IV.20: Square WINDOW
Figure IV.21: Graph of y x3 4x
To initialize the viewing window quickly to the standard viewing window (Figure IV.19), press F2 [Zoom] 6 [ZoomStd]. To set the viewing window quickly to a square window (Figure IV.20), press F[ZoomSqr]. More information about square windows is presented later in Section IV.2.4.
Figure IV.22: Custom WINDOW
Figure IV.23: Graph of y x3 4x
Sometimes you may wish to display grid points corresponding to tick marks on the axes. This and other graph format options may be changed while you are viewing the graph by pressing F1 to get the Tools menu (Figure IV.24) and then pressing 9 [Format] to display the Format menu (Figure IV.25) or by pressing | as indicated on the Tools menu in Figure IV.24. Move the blinking cursor to Grid; press 2 [ON] ENTER to redraw the graph. Figure IV.26 shows the same graph as in Figure IV.23 but with the grid turned on.

Figure IV.24: Tools menu

Figure IV.25: Format menu
Figure IV.26: Grid turned on for y x3 4x In general, youll want the grid turned off, so do that now by pressing OFF, then pressing ENTER.
and turning the Grid option to
IV.2.3 Graphing Step and Piecewise-Defined Functions: The greatest integer function, written x, gives the greatest integer less than or equal to a number x. On the TI-89, the greatest integer function is called floor( and is located under the Number sub-menu of the MATH menu (Figures IV.8-9). So, calculate 6.by pressing 2nd MATH [floor(] 6.78 ) ENTER. To graph y x, go into the Y= menu, move beside y1 and press CLEAR 2nd MATH X ) ENTER GRAPH. Figure IV.27 show this graph in a viewing window from 5 to 5 in both directions.
The true graph of the greatest integer function is a step graph, like the one in Figure IV.28. For the graph of y x, a segment should not be drawn between every pair of successive points. You can change this graph from a Line to a Dot graph on the TI-89 by going to the Y= screen, moving the cursor up until this function is selected (highlighted) and then pressing 2nd F6 [Style]. This opens the Graph Style menu. Move the cursor down to the second line and press ENTER or press 2; to have the selected graph plotted in Dot style. Now press GRAPH to see the result.
Figure IV.27: Line graph of y x
Figure IV.28: Dot graph of y x
Technology Tip: When graphing functions in the Dot style, it improves the appearance of the graph to set xres to 1. Figure IV.28 was graphed with xres 1. Also, the default graph style is Line, so you have to set the style to Dot each time you wish to graph a function in Dot mode. The TI-89 can graph piecewise-defined functions by using the when( function. The when( function is not on any of the keys but can be found in the CATALOG or typed from the keys. The format of the when( function is when(condition, trueResult, falseResult, unknownResult) where the falseResult and unknownResult are optional arguments. x2 2, x < 0 (using Dot graph), you want to graph xx 1, x 0 when the condition x < 0 is true and graph x 1 when the condition is false. First, clear any existing functions in the Y= screen. Then move to the y1 line and press 2nd a-lock W H E N alpha ( X 2nd < 0 , X ^ 2 + 2 , X 1 ) ENTER (Figure IV.29). Then press GRAPH to display the graph. Figure IV.30 shows this graph in a viewing window from 5 to 5 in both directions. This was done in Dot style, because the TI-89 will (incorrectly) connect the two sides of the graph at x 0 if the function is graphed in Line style. For example, to graph the function f x

Figure IV.29: Piecewise-defined function Other test functions, such as , , and the MATH menu.
Figure IV.30: Piecewise-defined graph
as well as logic operators can be found on the Test sub-menu of
IV.2.4 Graphing a Circle: Here is a useful technique for graphs that are not functions but can be split into a top part and a bottom part, or into multiple parts. Suppose you wish to graph the circle whose equation is x2 y2 36. First solve for y and get an equation for the top semicircle, y 36 x2, and for the bottom semicircle, y 36 x2. Then graph the two semicircles simultaneously. Use the following keystrokes to draw the circles graph. First clear any existing functions on the Y= screen. Enter 36 x2 as y1 and 36 x2 as y2 (see Figure IV.31) by pressing 2nd 36 X ^ 2 ) ENTER (-) 2nd 36 X ^ 2 ) ENTER. Then press GRAPH to draw them both (Figure IV.32).
Figure IV.31: Two semicircles
Figure IV.32: Circles graph standard WINDOW
If your range were set to the standard viewing window, your graph would look like Figure IV.32. Now this does not look like a circle, because the units along the axes are not the same. This is where the square viewing window is important. Press Fand see a graph that appears more circular. Technology Tip: Another way to get a square graph is to change the range variables so that the value of ymax ymin is approximately 38 times xmax xmin. For example, see the WINDOW in Figure IV.33 to 79 get the corresponding graph in Figure IV.34. This method works because the dimensions of the TI-89s display are such that the ratio of vertical to horizontal is approximately 38. 79

Figure IV.33:

vertical horizontal 39.5 79
Figure IV.34: A square circle
The two semicircles in Figure IV.34 do not connect because of an idiosyncrasy in the way the TI-89 plots a graph. Back when you entered 36 x2 as y2, you could have entered -y1 as y2 and saved some keystrokes. to move the cursor up to y2. Then press CLEAR (-) Try this by going into the Y= screen and pressing Y 1 ( X ) ENTER. The graph should be as before.
IV.2.5 Trace: Graph the function y x3 4x from Section IV.2.2 using the standard viewing window. (Remember to clear any other functions in the Y= screen.) Press any of the cursor directions and see the cursor move from the center of the viewing window. The coordinates of the cursors location are displayed at the bottom of the screen, as in Figure IV.35, in floating decimal format. This cursor is called a free-moving cursor because it can move from dot to dot anywhere in the graph window. Remove the free-moving cursor and its coordinates from the window by pressing GRAPH, CLEAR, ESC, or ENTER. Press the cursor directions again and the free-moving cursor will reappear at the same point you left it.
Figure IV.35: Free-moving cursor Press F3 [Trace] to enable the left and right directions to move the cursor from point to point along the graph of the function. The cursor is no longer free-moving, but is now constrained to the function. The coordinates that are displayed belong to points on the functions graph, so the y-coordinate is the calculated value of the function at the corresponding x-coordinate (Figure IV.36).

To center your window around a particular point, say h, k), and also have a certain x, set xmin h 79 x and make xmax h 79 x. Likewise, make ymin k 49 y and make ymax k 49 y. For example, to center a window around the origin 0, 0, with both horizontal and vertical increments of 0.25, set the range so that xmin 0.25 -19.75, xmax 0.25 19.75, ymin 0.25 12.25, and ymax 0.25 12.25. See the benefit by first graphing y x2 2x 1 in a standard viewing window. Trace near its y-intercept, which is 0, 1, and move towards its x-intercept, which is 1, 0. Then press F[ZoomDec] and trace again near the intercepts. IV.2.6 Zoom: Plot again the two graphs for y x3 4x and y .25x. There appears to be an intersection near x 2. The TI-89 provides several ways to enlarge the view around this point. You can change the viewing window directly by pressing WINDOW and editing the values of xmin, xmax, ymin, and ymax. Figure IV.42 shows a new viewing window for the range displayed in Figure IV.41. The cursor has been moved near the point of intersection; move your cursor closer to get the best approximation possible for the coordinates of the intersection.

Figure IV.41: New WINDOW

Figure IV.42: Closer view
A more efficient method for enlarging the view is to draw a new viewing window with the cursor. Start again with a graph of the two functions y x3 4x and y .25x in a standard viewing window (press Ffor the standard viewing window). Now imagine a small rectangular box around the intersection point, near x 2. Press F[ZoomBox] (Figure IV.43) to draw a box to define this new viewing window. Use the arrow keys to move the cursor, whose coordinates are displayed at the bottom of the window, to one corner of the new viewing window you imagine. Press ENTER to fix the corner where you moved the cursor; it changes shape and becomes a blinking square (Figure IV.44). Use the arrow keys again to move the cursor to the diagonally opposite corner of the new window (Figure IV.45). Note that you can press and hold or with or for this. If this box looks all right to you, press ENTER. The rectangular area you have enclosed will now enlarge to fill the graph window (Figure IV.46).

Figure IV.72: Angle measure Technology Tip: The automatic left parenthesis that the TI-89 places after functions such as sine, cosine, and tangent (as noted in Section IV.1.3) can affect the outcome of calculations. In the previous example, the degree sign must be inside of the parentheses so that when the TI-89 is in radian mode, it calculates the tangent of 45 degrees, rather than converting the tangent of 45 radians into an equivalent number of degrees. Also, the parentheses around the fraction are required so that when the TI-89 is in radian mode, it converts 6 into radians rather than converting merely the 6 to radians. Experiment with the placement of parentheses 6 to see how they affect the result of computation.
IV.4.2 Graphs of Trigonometric Functions: When you graph a trigonometric function, you need to pay careful attention to the choice of graph window and to your angle measure configuration. For example, graph sin 30x in the standard viewing window in radian mode. Trace along the curve to see where it is. Zoom 30 in to a better window, or use the period and amplitude to establish better WINDOW values. y Technology Tip: Because 3.1, when in radian mode, set xmin 0 and xmax 6.3 to cover the interval from 0 to 2. Next graph y tan x in the standard window first, then press F[ZoomTrig] to change to a special window for trigonometric functions in which the xscl is 1.5708 or 90 and the vertical range is from to 4. The TI-89 plots consecutive points and then connects them with a segment, so the graph is not exactly what you should expect. You may wish to change the plot style from Line to Dot (see Section IV.2.3) when you plot the tangent function.

Scatter Plots

IV.5.1 Entering Data: The table shows the total prize money (in millions of dollars) awarded at the Indianapolis 500 race from 1995 to 2003. (Source: Indy Racing League) Year Prize (in millions) 1995 $8.$8.$8.$8.$9.$9.$9.$10.$10.15
Well now use the TI-89 to construct a scatter plot that represents these points and to find a linear model that approximates the given data. The TI-89 holds data in lists. You can create as many list names as your TI-89 memory has space to store. Before entering this new data, clear the data in the lists that you want to use. To delete a list press 2nd VAR-LINK. This will display a list of folders showing the variables defined in each folder. Highlight the name of the list that you wish to delete and press F1 [Manage] 1 [Delete] ENTER. The TI-89 will ask you to confirm the deletion by pressing ENTER once more. Now press APPS 6 [Data/Matrix Editor] 3 [New.] P R I Z E ENTER to open a new variable called PRIZE (Figure IV.73). Press ENTER to then begin entering the variable values, with the years going in column c1. Instead of entering the full year, let x 5 represent 1995, x 6 represent 1996, and so on. Here are the keystrokes for the first three years: 5 ENTER 6 ENTER 7 ENTER and so on, then press to move to the next list. Move up to the first row and press 8.06 ENTER 8.11 ENTER 8.61 ENTER and so on (see Figure IV.74).

Figure IV.73: Entering a new variable
Figure IV.74: Entering data points
You may edit statistical data in almost the same way you edit expressions in the home screen. will delete the entire cell, not just the character or value to the left of the cursor. So, move the cursor to any value you wish to change, then type the correction. To insert or delete a data point, move the cursor over the data point (cell) you wish to add or delete. To insert a cell, move to the cell below the place where you want to insert the new cell and press 2nd F6 [Util ] 1 [Insert] 1 [cell ] and a new empty cell is open. IV.5.2 Plotting Data: First check the MODE screen (Figure IV.1) to make sure that you are in FUNCTION graphing mode. With the data points showing, press F2 [Plot Setup] to display the Plot Setup screen. If no other plots have been entered, Plot 1 is highlighted by default. Press F1 [Define] to select the options for the plot. Use , , and ENTER to select the Plot Type as Scatter and the Mark as a Box. Press alpha C 1 to set the independent variable, x, and press alpha C 2 to set the dependent variable, y, as shown in Figure IV.75, then press ENTER to save the options and press GRAPH to graph the data points. (Make sure that you have cleared or turned off any functions in the Y= screen, or those functions will be graphed simultaneously.) Figure IV.76 shows this plot in a window from 0 to 15 horizontally and vertically. You may now press F3 [Trace] to move from data point to data point.
Figure IV.75: Plot 1 menu
Figure IV.76: Scatter plot
To draw the scatter plot in a window adjusted automatically to include all the data you entered, press F[ZoomData]. When you no longer want to see the scatter plot press APPS [Current] F2, highlight Plot 1 and use F4 [] to deselect Plot 1 or press Y=, move the cursor up to highlight Plot 1, and press F4 []. The TI-89 still retains all the data you entered. IV.5.3 Regression Line: The TI-89 calculates slope and y-intercept for the line that best fits all the data. After the data points have been entered, while still in the Data/Matrix Editor, press F5 [Calc]. For the Calculation Type, choose 5 [LinReg] and set the x variable to c1 and the y variable to c2. In order to have the TI-89 graph the regression equation, set Store RegEQ to as y1(x) as shown in Figure IV.77. Press ENTER and the TI-89 will calculate a linear regression model with the slope named a and the y-intercept named b (Figure IV.78). The correlation coefficient (corr) measures how well the linear regression equation fits with the data. The closer the absolute value of the correlation coefficient is to 1, the better the fit; the closer the absolute value of the correlation coefficient is to 0, the worse the fit. The TI-89 displays both the correlation coefficient and the coefficient of determination R2.

Figure IV.84: Interchange rows 2 and 3
Figure IV.85: Add 4 times row 2 to row 3
To multiply row 2 by 4 and add the results to row 3, thereby replacing row 3 with new values, press 2nd MATH 4 alpha J 4 [mRowAdd( ] (-) 4 , alpha A , 2 , 3 ) ENTER (see Figure IV.85). The format of this command is mRowAdd(expression, matrix1, index1, index2). Technology Tip: Note that your TI-89 does not store a matrix obtained as the result of any row operations. So, when you need to perform several row operations in succession, it is a good idea to store the result of each one in a temporary place. x 2y 3z 9 For example, use row operations to solve this system of linear equations: x 3y 4. 2x 5y 5z 17
First enter this augmented matrix as a in your TI-89:
9 4. Then return to the home 17
screen and store this matrix as e (press alpha A STO alpha E ENTER) so you may keep the original in case you need to recall it. Here are the row operations and their associated keystrokes. At each step, the result is stored in e and replaces the previous matrix e. The last step of the row operations is shown in Figure IV.86. Row Operation Add row 1 to row 2. Add 2 times row 1 to row 3. Add row 2 to row 3. Multiply row 3 by 1. 2 Keystrokes 2nd MATH 4 alpha J 2 alpha E , 1 , 2 ) STO alpha E ENTER 2nd MATH 4 alpha J 4 (-) 2 , alpha E , 1 , 3 ) STO alpha E ENTER 2nd MATH 4 alpha J 2 alpha E , 2 , 3 ) STO alpha E ENTER 2nd MATH 4 alpha J 2 , alpha E , 3 ) STO alpha E ENTER
Figure IV.86: Row-echelon form of matrix after row operations So, z 2, y 1, and x 1.
Technology Tip: The TI-89 can produce a row-echelon form and the reduced row-echelon form of a matrix. The row-echelon form of matrix a is obtained by pressing 2nd MATH [ref( ] alpha A ) ENTER (see Figure IV.87) and the reduced row-echelon form is obtained by pressing 2nd MATH [rref(] alpha A ENTER (see Figure IV.88). Note that the row-echelon form of a matrix is not unique, so your calculator may not get exactly the same matrix as you do by using row operations. However, the matrix that the TI-89 produces will result in the same solution to the system.
Figure IV.87: Row-echelon form
Figure IV.88: Reduced row-echelon form
3 0. Because this consists 5 of the first three columns of the matrix a that was previously used, you can go to the matrix, move the cursor into the fourth column and press 2nd F6 [Util] 2 [Delete] 3 [column]. This will delete the column IV.6.4 Determinants and Inverses: Enter this matrix as a: 3 that the cursor is in. To calculate its determinant 0 , go to the home screen and press 2nd 5 MATH [det(] alpha A ) ENTER. You should find that the determinant is 2 as shown in Figure IV.89.

Figure IV.89: a and a1 Because the determinant of the matrix is not zero, it has an inverse matrix. Press alpha A ^ (-) 1 ENTER to calculate the inverse of matrix a. The result is shown in Figure IV.89.
Now lets solve a system of linear equations by matrix inversion. Once again, consider x 2y 3z x 3y 4. The coefficient matrix for this system is the matrix 0 which was 2x 5y 5z 9 entered as a in the previous example. Now enter the matrix 4 as b. Because b was used before, 17 when we stored 2a as b, press APPS [Open] 2 [Matrix] and use to move the cursor to b, then press ENTER twice to go to the matrix previously saved as b, which can be edited. Return to the home screen and press alpha A ^ (-) 1 alpha B ENTER to calculate the solution matrix (Figure IV.90). The solution is still x 1, y 1, and z 2.
Figure IV.90: Solution matrix

Sequences

IV.7.1 Iteration with the ANS key: The ANS feature enables you to perform iteration, the process of n1 n1 evaluating a function repeatedly. As an example, calculate for n 27. Then calculate for n the answer to the previous calculation. Continue to use each answer as n in the next calculation. Here are keystrokes to accomplish this iteration on the TI-89 calculator (see the results in Figure IV.91). Notice that when you use ANS in place of n in a formula, it is sufficient to press ENTER to continue an iteration. Iteration Keystrokes 27 ENTER ( 2nd ANS 1 ) 3 ENTER ENTER ENTER Display 27 8.666666667 2.555555556.5185185185
Figure IV.91: Iteration Press ENTER several more times and see what happens with this iteration. You may wish to try it again with a different starting value. IV-32
IV.7.2 Terms of Sequences: Another way to display the terms of a sequence is to enter the sequence and the number of terms you want listed. For example, to find the first five terms of the sequence un n 4, press 2nd MATH 3 [List] 1 [seq( ] (-) alpha N + 4 , alpha N , 1 , 5 , 1 ) ENTER (see Figure IV.92). The format of this command is seq(expression, variable, low, high, step).
Figure IV.92: Terms of sequence un n 4 IV.7.3 Arithmetic and Geometric Sequences: Use iteration with the ANS variable to determine the nth term of a sequence. For example, find the 18th term of an arithmetic sequence whose first term is 7 and whose common difference is 4. Enter the first term 7, then start the progression with the recursion formula, 2nd ANS + 4 ENTER. This yields the 2nd term, so press ENTER sixteen more times to find the 18th term, 75. For a geometric sequence whose common ratio is 4, start the progression with 2nd ANS 4 ENTER. You can also define the sequence recursively with the TI-89 by selecting SEQUENCE in the Graph type on the first page of the MODE menu (see Figure IV.1). Once again, lets find the 18th term of an arithmetic sequence whose first term is 7 and whose common difference is 4. Press MODE 4 [SEQUENCE] ENTER. Y= to edit any of the TI-89s sequences, u1 through u99. Make u1 u1n and Then press uiby pressing alpha U 1 ( alpha N 1 ) + 4 ENTER 7 ENTER (Figure IV.93). Press 2nd QUIT to return to the home screen. To find the 18th term of this sequence, calculate u1(18) by pressing alpha U 1 ( 18 ) ENTER (see Figure IV.94).

Figure IV.93: Sequence Y= menu
Figure IV.94: Sequence mode
Of course, you could also use the explicit formula for the nth term of an arithmetic sequence, tn a n 1d. First enter values for the variables a, d, and n, then evaluate the formula by pressing alpha A + ( alpha N 1 ) alpha D ENTER. For a geometric sequence whose nth term is given by tn a r n1, enter values for the variables a, n, and r, then evaluate the formula by pressing alpha A alpha R ^ ( alpha N 1 ) ENTER. To use the explicit formula in sequence mode, make un by pressing Y= then using to move up to the u1 line and pressing CLEAR 7 + ( N 1 ) 4 ENTER 2nd QUIT. Once more, calculate u1(18) by pressing alpha U 1 ( 18 ) ENTER.
IV.7.4 Finding Sums and Partial Sums of Sequences: You can find the sum of a sequence by combining the sum( feature with the seq( feature on the List sub-menu of the MATH menu. The format of the sum( command is sum(list). The format of the seq( command is seq(expression, variable, low, high, step) where the step argument is optional and the default is for integer values from low to high. For example, suppose you want to find the sum
40.3. Press 2nd MATH [sum( ] 2nd MATH 4
(. 3 ) ^ alpha N , alpha N , 1 , 12 ) ) ENTER (Figure IV.95). The seq( command generates a list, which the sum( command then sums. Note that any letter can be used for the variable in the sum, i.e., the N could just have easily been an A or a K.
Figure IV.95: Now calculate the sum starting at n 0 by using of approximately 5.71284803. ,
to edit the range. You should obtain a sum
The seq( feature can also be combined with the cumSum( feature to find partial sums of a series. The IV-33 format of the cumSum( command is cumSum(list). For example, suppose you want to find the first four partial sums of the series

Press 2nd MATH

[cumSum( ] 2nd MATH 3 ^ ( alpha N + 1) , alpha N , 1 , 4 ) ) ENTER (Figure IV.96).
Figure IV.96: Partial sums of
Parametric and Polar Graphs
IV.8.1 Graphing Parametric Equations: The TI-89 plots up to 99 pairs of parametric equations as easily as it plots functions. In the first page of the MODE menu (Figure IV.1) change the Graph setting to PARAMETRIC. Be sure, if the independent parameter is an angle measure, that the angle measure in the MODE menu is set to whichever you need, RADIAN or DEGREE.

Figure IV.105: Entering a new variable
Figure IV.106: List editor
Figure IV.107: Sum IV.9.5 Statistics: The following data are the high temperatures (in degrees Fahrenheit) recorded in Lincoln, Nebraska from October 1, 2003 to October 12, 2003 (Source: University of Nebraska-Lincoln) 65, 68, 74, 79, 83, 81, 80, 80, 79, 72, 67, 71 To find the mean and median of these temperatures, first enter the data using the TI-89s list editor. Press APPS 3 T E M P S ENTER ENTER to open a new variable called TEMPS (see Figure IV.108). Now begin entering the temperatures as shown in Figure IV.109. Then press 2nd QUIT. To find the mean, press 2nd MATH 6 [Statistics] 4 [mean( ] 2nd a-lock T E M P S alpha ) ENTER and to find the median, press 2nd MATH [median( ] 2nd a-lock T E M P S alpha ) ENTER (see Figure IV.110). So, the mean of the temperatures is approximately 75F and the median is 76.5F.
Figure IV.108: Entering a new variable
Figure IV.109: List editor
Figure IV.110: Mean and median You can also find the mean and median of the above data by using the OneVar command found in the Calc menu of the Data/Matrix Editor. You can copy the data you entered in Figure IV.108 to a data list by opening the TEMPS list first. Then press F[Save Copy As] 1 [Data] T E M P S alpha 2 ENTER ENTER (see Figure V.109). Note that you cannot name the data list TEMPS. Now, to use the OneVar command you must have the data list TEMPS2 open. Then press F5 [Calc]. For the Calculation Type, choose 1 [OneVar], set the x variable to c1, and press ENTER ENTER. The TI-89 will calculate several different statistical values. The first line represents the mean of the data which is approximately 75F (see Figure IV.112). The second line is the sum of the data, the third line is the sum of the squares of the data, the IV-38
fourth line is the sample standard deviation of the data, the fifth line is the number of data values, the sixth line is the minimum value of the data, the seventh line is the first quartile of the data, and the eighth line is the median of the data which is 76.5F (see Figure IV.113). The ninth line is the third quartile of the data and the tenth line is the maximum value of the data.
Figure IV.111: Saving a list as a data list
Figure IV.112: OneVar command
Figure IV.113: OneVar command You can scroll through the list of statistical values by pressing or.

IV.10 Programming

IV.10.1 Entering a Program: The TI-89 is a programmable calculator that can store sequences of commands for later replay. Press APPS 7 [Program Editor] to access the programming menu. The TI-89 has space for many programs, each called by a title you give it. To create a new program, start by pressing APPS [New]. Set the Type to Program and the Folder to main (unless you have another folder in which you want to store the program). Enter a descriptive title for the program in the Variable line. After you name the program, press ENTER ENTER to go to the program editor. The program name and the beginning and ending commands of the program are automatically displayed with the cursor on the first line after Prgm, the begin program command. In the program, each line begins with a colon : supplied automatically by the calculator. Any command you could enter directly in the TI-89s home screen can be entered as a line in a program. There are also special programming commands. You may interrupt programming input at any stage by pressing 2nd QUIT. To return later for more editing, press APPS [Open], move the cursor down to the Variable list, highlight the programs name, and press ENTER ENTER. You may remove a program from memory by pressing 2nd VAR-LINK, move the cursor to highlight the name of the program you want to delete, then press F1 [Manage] 1 [Delete] ENTER and then ENTER again to confirm the deletion from the calculators memory.

Part III: III.1

Texas Instruments TI-89 Graphics Calculator
Systems of Linear Equations
III.1.1 Basics: Press the ON key to begin using your TI-89 calculator. If you need to adjust the display contrast, first press , then press (the minus key) to lighten or (the plus key) to darken. To lighten or darken the screen more, press then or again. When you have finished with the calculator, turn it off to conserve battery power by pressing 2nd and then OFF. Check the TI-89s settings by pressing MODE. If necessary, use the arrow key to move the blinking cursor to a setting you want to change. You can use F1 to go to page 1, F2 to go to page 2, or F3 to go to page 3 of the MODE menu. To change a setting, use to get to the setting that you want to change, then press to see the options available. Use the or to highlight the setting that you want and press ENTER to select the setting. To start, select the options shown in Figures III.1, III.2, and III.3: function graphs, main folder, floating decimals with 10 digits displayed, radian measure, normal exponential format, real numbers, rectangular vectors, pretty print, full screen display, home screen showing, approximate calculation, base 10 system, International System of Units, and English language. Note that some of the lines of the MODE menu are not readable. These lines pertain to options that are not set above. For now, leave the MODE options by pressing HOME or 2nd QUIT. Some of the current settings are shown on the status line of the Home screen.
Figure III.1: MODE menu, page 1
Figure III.2: MODE menu, page 2
Figure III.3: MODE menu, page 3

Figure III.4: APPS menu

Technology Tip: There are many ways to get the most commonly used screens on your TI-89. One method is by using APPS menu (see Figure III.4) which is accessed by pressing the blue APPS key. To return to the Home screen press 2nd QUIT, HOME, or ENTER. III.1.2 Key Functions: Most keys on the TI-89 offer access to more than one function, just as the keys on a computer keyboard can produce more than one letter (g and G) or even quite different characters (5 and %). The primary function of a key is indicated on the key itself, and you access that function by a simple press on the key. To access the second function indicated to the left above a key, first press 2nd (2nd appears on the status line) and then press the key. For example, to calculate 25, press 2nd 25 ) ENTER. Technology Tip: There are separate keys for the commonly used letters x, y, z, and t. A simple press of the key will produce a lowercase letter while pressing and the key will produce an uppercase letter. TI-89 Graphics Calculator
Copyright by Houghton Mifflin Company. All rights reserved.
NN after you have entered a value for N. Suppose you want N = 200. Press 200 STO N ENTER to store the value 200 in memory location N. Whenever you use N in an expression, the calculator will substitute the NN 1 value 200 until you make a change by storing another number in N. Next enter the expression by 2 NN 1 20100. typing Nx( N + 1 ) 2 ENTER. For N 200, you will find that 2 III.1.3 Algebraic Expressions and Memory: Your calculator can evaluate expressions such as Technology Tip: The contents of any memory location may be revealed by typing just its letter name and then ENTER. Simply press N ENTER to see the current value of the variable N. And the TI-89 retains memorized values even when it is turned off, so long as its batteries are good. Technology Tip: Because variable names may be more than one character in length, multiplication between variables must always be expressed. So for the product ab, you must enter alpha a x alpha b with the multiplication key. With a numerical coefficient, however, the multiplication does not need to be expressed; hence for 4ab you may enter 4 alpha a x alpha b. III.1.4 The MATH Menu: Operators and functions associated with a scientific calculator are available either immediately from the keys of the TI-89, by 2nd keys, or by keys. You have direct key access to common arithmetic operations (2nd , ^), trigonometric functions (2nd SIN, 2nd COS, 2nd TAN) and their inverses ( SIN1, COS1, TAN1), exponential and logarithmic functions (2nd LN, x), and a famous constant (2nd ). e Note that the TI-89 distinguishes between subtraction and the negative sign. If you wish to enter a negative number, it is necessary to use the (-) key. For example, you would evaluate 3 by pressing (-) 5 ( 4 x (-) 3 ) ENTER to get 7.
Figure III.5: Subtraction and the Negative sign A significant difference between the TI-89 and many scientific calculators is that the TI-89 requires the argument of a function after the function, as you would see a formula written in your textbook. For example, on the TI-89 you calculate 16 by pressing the keys 2nd 16 ) in that order. Here are keystrokes for basic mathematical operations. Try them for practice on your TI-89. Expressions

Keystrokes 2nd 3 ^ 2 + 4 ^ 2) ENTER 2 + 3 ^ (-) 1 ENTER or 2 + (1 3) ENTER 2nd LN 200 ) ENTER 2.34 x 10 ^ 5 ENTER
Display 5. 2.333333333 5.298317367 234000.

ln 200 2.34 105

TI-89 Graphics Calculator
Technology Tip: Note that if you had set the calculation to either AUTO or EXACT (second to the last line of page 2 of the MODE menu), the TI-89 would display 3 for 23 and 2 ln5 ln2 for ln 200. Thus, you can use either fractions and exact numbers or decimal approximations. The AUTO mode will give exact rational results whenever all of the numbers entered are rational, and decimal approximations for other results. Additional mathematical operations and functions are available from the MATH menu. Press 2nd MATH to see the various sub-menus. Press 1 [Number] or just ENTER to see the options available under the Number sub-menu. You will learn in your mathematics textbook how to apply many of them. As an example, calculate the remainder of 437 when divided by 49 by pressing 2nd MATH 1 [Number] then either alpha a [remain (] or ENTER; finally press 437 , 49 ) ENTER to see 45. To leave the MATH menu (or any other menu) and take no other action, press 2nd QUIT or just ESC.
Figure III.6: MATH Number menu
Figure III.7: remain ( function
Note that you can select a function or a sub-menu from the current menu by pressing either until the desired item is highlighted and then ENTER, or by pressing the number or letter corresponding to the function or sub-menu. It is easier to press alpha a than to press nine times to get the remain( function. The factorial of a non-negative integer is the product of all the integers from 1 up to the given integer. The symbol for factorial is the exclamation point. So 4! (pronounced four factorial) is 24. You will learn more about applications of factorials in your textbook, but for now use the TI-89 to calculate 4! Press the keystrokes: 4 2nd MATH 7 [Probability] ENTER ENTER. III.1.5 Graphing Linear Functions: Once you have entered a function in the Y= screen of the TI-89, just press GRAPH to see its graph. Y= key (above the F1 key) or APPS 2 [Y= For example, here is how to graph y x 3. Press the Editor] to display the function editing screen (Figure III.8). You may enter as many as ninety-nine different functions for the TI-89 to use at one time. If there is already a function y1 press or as many times as necessary to move the cursor to y1 and then press CLEAR to delete whatever was there. Then enter the expression x 3 by pressing (-) x + 3 ENTER. Now press GRAPH (above the F3 key).

Figure III.8: Y= screen

Figure III.9: Graph of y x 3
Technology Tip: While the TI-89 is calculating coordinates for a plot, it displays the word BUSY on the status line. The viewing rectangle in Figure III.9 shows the part of the graph that extends horizontally from 10 to 10 and vertically from 10 to 10. Press WINDOW to see information about your viewing rectangle. Figure III.10 shows the WINDOW screen that corresponds to the viewing rectangle in Figure III.9. This is the standard viewing rectangle for the TI-89.

Figure III.10: Standard WINDOW The variables Xmin and Xmax are the minimum and maximum x-values of the viewing rectangle: Ymin and Ymax are its minimum and maximum y-values. Xscl and Yscl set the spacing between tick marks on the axes. Xres sets the resolutions. Use the arrow keys and to move up and down from one line to another in this list; pressing the ENTER key will move down the list. Press CLEAR to delete the current value and then enter a new value. Remember that a minimum must be less than the corresponding maximum or the TI-89 will issue an error message. Also, remember to use the (-) key, not (which is subtraction), when you want to enter a negative value. Technology Tip: To set the range quickly to standard values (see Figure III.10), press [Zoom] 6 [ZoomStd]. WINDOW F2
Technology Tip: If you would like to see a function in the Y = menu and its graph in a graph window, both at the same time, press MODE to open the MODE menu and press F2 to go to the second page. The cursor will be next to Split Screen. Select either TOP-BOTTOM or LEFT-RIGHT by pressing and 2 or 3, respectively. Now the 2 lines below the Split 1 APP line have become readable, since these options apply only when the calculator is in the split screen mode. The Split 1 APP will automatically be the screen you were on prior to pressing MODE. You can choose what you want the top or left-hand screen to show by moving down to the Split 1 APP line, pressing and the number of the application you want in that window. The Split 2 APP determines what is shown in the bottom or right-hand window. Press ENTER to confirm your choices and your TI-89s screen will now be divided either horizontally or vertically (as you choose). Figure III.11 shows the graph and the Y = screen with the settings shown in Figure III.12. The split screen is also useful when you need to do some calculations as you trace along a graph. In split screen mode, one side of the screen will be more heavily outlined. This is the active screen, i.e., the screen that you can currently modify. You can change which side is active by using 2nd to access the symbol above the APPS key. For now, restore the TI-89 to Full screen.

Figure III.11: Split Screen
Figure III.12: Settings for split screen
III.1.6 Graphing Parametric Functions: The TI-89 plots parametric equations as easily as it plots functions. Up to ninety-nine pairs of parametric equations can be plotted. In the first page of the MODE menu (Figure III.1) change the GRAPH setting to PARAMETRIC. Be sure, if the independent parameter is an angle measure, that the angle measure in the MODE menu has been set to whichever you need, RADIAN or DEGREE. You can now enter the parametric functions. For example, here are the keystrokes needed to graph the parametric equations x cos3 t and y sin3 t. First check that angle measure is in radians. Then press Y= ( 2nd COS T ) ) ^ 3 ENTER ( 2nd SIN T ) ) ^ 3 ENTER (Figure III.13). Press WINDOW to set the graphing window and to initialize the values of t. In the standard window, the values of t go from 0 to 2 in steps of 0.1309, with the view from 10 to 10 in both directions. In 24 order to provide a better viewing rectangle press ENTER three times and set the rectangle to go from 2 to 2 horizontally and vertically (Figure III.14). Now press GRAPH to draw the graph (Figure III.15).
Figure III.13: Parametric Y = menu
Figure III.14: Parametric WINDOW menu
Figure III.15: Parametric graph of x cos3 t and y sin3 t III.1.7 Solving Linear Systems: The solutions to a system of equations correspond to the points of inter3x y 1 section of their graph. As an example, lets graph and solve the system. 2x y 0
First transform each equation by solving for y: and 2x for Y2 (Figure III.16).

y 3x 1. Then press y 2x

Y = and enter 3x 1 for Y1
Figure III.16: 3x 1 for Y1 and 2x for Y2
Figure III.17: Setting up to locate the intersection
Find the coordinates of a point of intersection of the two graphs by pressing GRAPH F5 5. Trace with the cursor keys or first along one graph near an intersection (Figure III.17) and press ENTER; then trace with the cursor along the other graph and press ENTER. Move the cursor just left of the point of intersection (Figure III.18) and press ENTER again. Finally, move the cursor just right of the point of intersection and press ENTER again. Coordinates of the intersection will be displayed at the bottom of the window (Figure III.19).
Figure III.18: Setting up to locate the intersection
Figure III.19: Point of intersection
The TI-89 also has a solve( function that you can use to solve a linear system. The technique is based on the y 3x 1 fact that any solution of the system is a root of the equation 3x 1 2x. So press For y 2x (2nd MATH 9 [Algebra] 1) 3x 1 = 2x , x ) then press ENTER for the x-coordinate of the point of intersection (Figure III.20). Then to calculate its y-coordinate, save this value as x (press 1 STO x ENTER) and evaluate either 3x 1 or 2x.

Figure III.20: Solve ( function
Figure III.21: Calculate the y-coordinate

Matrices

III.2.1 Making a Matrix: The TI-89 can work with as many different matrices as the memory will hold. Heres how to create the matrix in your calculator. 5 17
From the Home screen, press APPS 6 [Data/Matrix Editor] 3 [New]. Set the Type to Matrix, the Variable to a (this is the name of the matrix), the Row Dimension to 3 and the Col Dimension to 4 (Figure III.22). Press ENTER to accept these values.
Figure III.22: Data/Matrix menu
Figure III.23: Editing a matrix
The display will show the matrix by showing a grid with zeros in the rows and columns specified in the definition of the matrix. Use the cursor pad or ENTER repeatedly to move the cursor to a matrix element you want to change. If you press ENTER, you will move right across a row and then back to the first column of the next row. The lower left of the screen shows the cursors current location within the matrix. The element in the second row and first column in Figure III.23 is highlighted, so the lower left of the window is r2c1 1, showing that elements current value. Enter all the elements of matrix a; pressing ENTER after inputting each value. When you are finished, leave the editing screen by pressing 2nd QUIT or HOME to return to the home screen. Technology Tip: The TI-89 enables you to create an identity matrix quickly. If you want to make the identity matrix, for example, press 2nd MATH 4 [Matrix] 6 [identity] 3 ) ENTER (see Figure III.24). If you want to save the identity matrix as matrix b, press 2nd Math 4 [Matrix] 6 [identity] 4 ) STO alpha b ENTER. Technology Tip: The TI-89 also enables you to create a matrix of any size and fill it with random singledigit integers 9 to 9. To create a matrix filled with random integers, press 2nd MATH 4 [Matrix] alpha e [randMat(] 2 , 3 ) ENTER (see Figure III.25).
Figure III.24: Identity matrix
Figure III.25: Random matrix
From the Home screen, you can perform many calculations with matrices. To see matrix a, press alpha a ENTER. TI-89 Graphics Calculator
III.2.2 Scalar Multiplication: Perform the scalar multiplication 2a pressing 2 alpha a ENTER. The resulting matrix is displayed on the screen. To create matrix b as 2a press 2 alpha a STO alpha b ENTER (Figure III.27), or if you do this immediately after calculating 2a, press only STO alpha b ENTER. The calculator will display the matrix.

Figure III.26: Matrix a

Figure III.27: Matrix b
III.2.3 Matrix Addition: To add two matrices, say a and b, create b (with the same dimensions as a) and then press alpha a + alpha b ENTER. Again, if you want to store the answer as a specific matrix, say m, then press STO alpha m. Subtraction is performed in a similar manner. III.2.4 Matrix Multiplication: Now create a matrix called c with dimensions of 2 3. Enter the matrix 3 as c. For matrix multiplication of c by a, press alpha c alpha a ENTER. If you tried to 1 multiply a by c, your TI-89 would notify you of an error because the dimensions of the two matrices do not permit multiplication in this way.
Figure III.28: Matrix multiplication

Figure III.29: Transpose

III.2.5 Transpose of a Matrix: The transpose of a matrix is another matrix with the rows and columns interchanged. The symbol for the transpose of a is aT. To calculate aT, press alpha a 2nd MATH 4 [Matrix] 1 [T] ENTER (see Figure III.29). III.2.6 Row Operations: Here are the keystrokes necessary to perform elementary row operations on a matrix. Your textbook provides more careful explanation of the elementary row operations and their uses. To interchange the second and third rows of the matrix a that was defined above, press 2nd MATH 4 [Matrix] alpha j [Row ops] 1 [rowSwap(] alpha a , 2 , 3 ) ENTER (see Figure III.30). The format of this command is rowSwap(matrix, row1, row2).
Figure III.30: Swap rows 2 and 3
Figure III.31: Add -4 times row 2 to row 3
To add row 2 and row 3 and store the results in row 3, press 2nd MATH 4 [Matrix] alpha j [Row ops] 2 [rowAdd(] alpha a , 2 , 3 ) ENTER. The format of this command is rowAdd(matrix, row1, row2). To multiply row 2 by 4 and store the results in row 2, thereby replacing row 2 with new values, press 2nd MATH 4 [Matrix] alpha j [Row ops] 3 [mRow(] (-) 4 , alpha a , 2 ) ENTER. The format of this command is mRow(expression, matrix1, index). To multiply row 2 by 4 and add the results to row 3, thereby replacing row 3 with new values, press 2nd MATH 4 [Matrix] alpha j [Row ops] 4 [mRowAdd(] (-) 4 , alpha a , 2 , 3 ) ENTER (see Figure III.31). The format of this command is mRowAdd(expression, matrix1, index1, index2). Note that your TI-89 does not store a matrix obtained as the result of any row operation. So when you need to perform several row operations in succession, it is a good idea to store the result of each one in a temporary place. x 2y 3z 9 For example, use elementary row operations to solve this system of linear equations: x 3y 4. 2x 5y 5z 9 First enter this augmented matrix as a in your TI-89: 0 4. Then return to the Home screen and store this matrix as e (press alpha a STO alpha e ENTER), so you may keep the original in case you need to recall it. Here are the row operations and their associated keystrokes. At each step, the result is stored in e and replaces the previous matrix e. The last two steps of the row operations are shown in Figure III.32. Row Operation add row 1 to row 2 Keystrokes 2nd MATH 4 alpha j 2 alpha e , 1 , 2 ) STO alpha e ENTER

add 2 times row 1 to row 3 2nd MATH 4 alpha j 4 (-) 2 , alpha e , 1 , 3 ) STO alpha e ENTER add row 2 to row 3 multiply row 3 by
2nd MATH 4 alpha j 2 alpha e , 2 , 3 ) STO alpha e ENTER 2nd MATH 4 alpha j 2 , alpha e , 3 ) STO alpha e ENTER
Figure III.32: Row operations Thus z 2, so y 1, and x 1. Technology Tip: The TI-89 can produce a row-echelon form and the reduced row-echelon form of a matrix. The row-echelon form of matrix a is obtained by pressing 2nd MATH 4 [Matrix] 3 [ref(] alpha a ) ENTER and the reduced row-echelon form is obtained by pressing 2nd MATH 4 [Matrix] 4 [rref(] alpha a ) ENTER. Note that the row-echelon form of a matrix is not unique, so your calculator may not get exactly the same matrix as you do by using row operations. However, the matrix that the TI-89 produces will result in the same solution to the system. 3 III.2.7 Determinants and Inverses: Enter the matrix as a: 0. Since this consists of the 5 first three columns of the matrix a that was previously used, you can go to the matrix, move the cursor into the fourth column and press F6 [Util] 2 [Delete] 3 [column]. This will delete the column that the cursor is 3 in. To calculate its determinate, 0 , go to the Home screen and press 2nd MATH 4 [Matrix] [det(] alpha a ) ENTER. You should find that the determinant is 2 as shown in Figure III.33.
Figure III.33: Determinant of a
Figure III.34: Inverse of a
Since the determinant of the matrix is not zero, it has an inverse matrix. Press alpha a ^ (-) 1 ENTER to calculate the inverse. The result is shown in Figure III.34. Now lets solve a system of linear equations by matrix inversion. Once again, consider x 2y 3z x 3y 4. The coefficient matrix for this system is the matrix 0 which was entered 2x 5y 5z 5 5
9 as a matrix a in the previous example. Now enter the matrix 4 as b. Since b was used before, when we 17 stored 2a as b, press APPS 6 [Data/Matrix Editor] 2 [Open] III-10

2 [Matrix]

and use

to move

the cursor to b, then press ENTER twice to go to the matrix previously saved as b, which can be edited. Return to the Home screen and press alpha a ^ (-) 1 alpha b ENTER to get the answer as shown in Figure III.35.
Figure III.35: Solution matrix The solution is still x 1, y 1, and z 2. 3 III.2.8 LU-Factorization: Use the square matrix a: 0. To calculate its LU-factorization, 5 press 2nd MATH 4 [Matrix] alpha b [LU] alpha a , alpha l , alpha u , alpha p ENTER. The format of this command is LU(matrix, lower triangular matrix, upper triangular matrix, permutation matrix).
Figure III.36: LU-factorization and the lower triangular matrix 1 III.2.9 Eigenvalues: Enter the square matrix a: 0. Calculate the eigenvalues of matrix a by 1 pressing 2nd MATH 4 [Matrix] 9 [eigVl(] alpha a ) ENTER. Your TI-89 returns a list of eigenvalues of a real or complex square matrix (Figure III.37).

Figure III.37: Eigenvalues and Eigenvector

III-11

1 III.2.10 Eigenvectors: Calculate the eigenvectors of matrix a = 0 by pressing 2nd MATH [Matrix] alpha a [eigVc(] alpha a ) ENTER. The calculator returns a matrix, each column of which is an eigenvector corresponding to an eigenvalue (Figure III.37). Technology Tip: The entry 1. E -15 in Figure III.37 (third row, second column of the eigenvector matrix) should be taken as essentially 0.

Additional Topics

III.3.1 Length and Dot Product in Rn: Create a vector in the TI-89 with square brackets. For example, make the vector V = (0, 2, 1, 4, 2) by pressing 2nd [ 0 , (-) 2 , 1 , 4 , (-) 2 2nd ] STO V ENTER.
Figure III.38: Vector norm and dot product To calculate the length (norm) of the vector press 2nd MATH 4 [Matrix] alpha h [Norms] 1 [norm(] alpha v ) ENTER (see Figure III.38). Now define the two vectors U 1, 2, 0, 3 and V 3, 2, 4, 2. Calculate the dot product U V of the vectors by pressing 2nd MATH 4 [Matrix] alpha l [Vector ops] 3 [dotP (] U, V ) ENTER (see Figure III.38). III.3.2 Cross Products: Evaluate the cross product of two vectors, u i 2j k and v 3i j 2k, by pressing 2nd MATH 4 [Matrix] alpha l [Vector ops] 2 [crossP(] 2nd [ 1 , (-) 2 , 1 2nd ] , 2nd [ 3 , 1 , (-) 2 2nd ] ) ENTER. The cross product is u v = 3i + 5j + 7k.
Figure III.39: Vector cross products III.3.3 Complex Numbers: Press 2nd MATH 5 [Complex] to display the menu of special complex number operators (Figure III.40).

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Expressions
Keystrokes ( 2 + 3 2nd i ) + ( 2nd i ) ENTER ( 2 + 3 2nd i ) x ( 2nd i ) ENTER ( 31 + 2nd i ) ( 2nd i ) ENTER 2nd MATH 5 [Complex] 5 [abs(] 2nd i ) ENTER 2nd MATH 5 [Complex] 1 [conj(] 2nd i ) ENTER
Answer 7 4i 31 i 2 3i 12i

2 3i 5 7i 2 3i5 7i

31 i 5 7i
Figure III.40: Complex number arithmetic III.3.4 Rectangular-Polar Conversion: The ANGLE sub-menu of the MATH menu provides a function for converting between rectangular and polar coordinate systems. These functions use the current angle measure setting, so it is a good idea to check the default angle measure before any conversion. For the following examples, the TI-89 is set to radian measure. Given the rectangular coordinates x, y 4, 3, convert to polar coordinates r, in the Home screen by pressing 2nd MATH 2 [ANGLE] 5 [R Pr(] 4 , (-) 3 ) ENTER to display the value of r. The value of is displayed after you press 2nd MATH 2 [ANGLE] 6 [R P(] 4 , (-) 3 ) ENTER. The polar coordinates are approximately 5, 0.6435. Suppose r, 3,. Convert to rectangular coordinates x, y by pressing 2nd MATH 2 [ANGLE] 3 [P Rx(] 3 , 2nd ) ENTER. The x-coordinates displayed. Press 2nd MATH 2 [ANGLE] 4 [P Ry(] 3 , 2nd ) ENTER to display the y-coordinate (Figure III.41). The rectangular coordinates are 3, 0.

Figure III.41: Converting between rectangular and polar coordinates
Program: Visualizing Row Operations
III.4.1 Entering the Program: The TI-89 is a programmable calculator that can store sequences of commands for later replay. Heres a useful program that demonstrates how elementary matrix row operations used in Gauss-Jordan elimination may be interpreted graphically.

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Press APPS 7 [Program Editor] to access the programming menu. The TI-89 has space for many programs, each called by a name you give it. Create a new program, so press APPS 7 [Program Editor] 3 [New]. Set the Type to Program and the Folder to main (unless you have another folder in which you want to have the program). Enter a descriptive title for the program in the Variable line. Name this program Rowops and press ENTER twice to go to the program editor. The program name and the beginning and ending commands of the program are automatically displayed with the cursor on the first line after Prgm, the begin program command. In the program, each line begins with a colon : supplied automatically by the calculator. Any command you could enter directly in the TI-89s Home screen can be entered as a line in a program. There are also special programming commands.
Figure III.42: Part of program: ROWOPS Input the program ROWOPS by pressing the keystrokes given in the following listing. You may interrupt program input at any stage by pressing 2nd QUIT. To return later for more editing, press APPS 7 [Program Editor] 2 [Open], move the cursor down to Variable list, highlight the programs name, and press ENTER twice. Program Line :rowops () :Prgm :Clrio: ClrGraph CATALOG C [arrow down to ClrIO] ENTER 2nd : CATALOG C [arrow down to ClrGraph] ENTER ENTER Keystrokes
:Disp ENTER A 2 BY 3 MATRIX F[Disp] 2nd a-lock E N T E R A alpha 2 a-lock BY alpha 3 a-lock M A T R I X 2nd ENTER :Disp a b c :Disp d e f F[Disp] 2nd A F[Disp] 2nd D B E C 2nd ENTER F 2nd ENTER
:Prompt a, b, c, d, e, f F[Prompt] alpha A , alpha B , alpha C , alpha D , alpha E , alpha F ENTER :[[a, b, c][d, e, f]]m 2nd [ 2nd [ alpha A , alpha B , alpha C 2nd ] 2nd [ alpha D , alpha E , alpha F 2nd ] 2nd ] STO alpha M ENTER :ClrIO CATALOG C [arrow down to ClrIO] ENTER ENTER a-lock O R I G I N A L MATR I X
:Disp ORIGINAL MATRIX F[Disp] 2nd 2nd ENTER III-14

:Pause m :b1 (ca*x)y2 (x) :e1 (fd*x)y1 (x)
CATALOG P [arrow down to Pause] ENTER alpha M ENTER alpha B ^ (-) 1 ( alpha C alpha A X ) STO Y 2 ( X ) ENTER alpha E ^ (-) 1 ( alpha F alpha D X ) STO Y 1 ( X ) ENTER
:ZoomStd: Pause: ClrIO CATALOG Z [arrow down to ZoomStd] ENTER 2nd : CATALOG P [arrow down to Pause] ENTER 2nd : CATALOG C [arrow down to ClrIO] ENTER ENTER :Disp OBTAIN LEADING F[Disp] 2nd 2nd ENTER :Disp 1 IN ROW 1 :mRow (a1, m, 1)m :Pause m: ClrDraw :(a/b) (c/ax)y2(x) :DispG: Pause: ClrIO a-lock O B T A I N a-lock IN LEADING ROW
F[Disp] 2nd alpha 1 alpha 1 2nd ENTER
2nd MATH 4 [Matrix] alpha J [Row ops] 3 [mRow(] alpha A ^ (-) 1 , alpha M , 1 ) STO alpha M ENTER CATALOG P [arrow down to Pause] ENTER alpha M 2nd : CATALOG C [arrow down to ClrDraw] ENTER ENTER ( alpha A alpha B ) ( alpha C alpha A X ) STO Y 2 ( X ) ENTER CATALOG D [arrow down to DispG] ENTER 2nd : CATALOG P [arrow down to Pause] ENTER 2nd : CATALOG C [arrow down to ClrIO] ENTER ENTER alpha 0 a-lock a-
:Disp OBTAIN 0 BELOW F[Disp] 2nd a-lock O B T A I N lock B E L O W 2nd ENTER :Disp LEADING 1 IN :Disp COLUMN 1 F[Disp] 2nd L E A D I N G I N 2nd ENTER F[Disp] 2nd C O L U M N alpha 1

alpha 1 2nd ENTER

:mRowAdd (d, m, 1, 2)m 2nd MATH 4 [Matrix] J [Row ops] 4 [mRowAdd(] (-) alpha D , alpha M , 1 , 2 ) STO alpha M ENTER :Pause m: ClrDraw CATALOG P [arrow down to Pause] ENTER alpha M 2nd : CATALOG C [arrow down to ClrDraw] ENTER ENTER
:(eb*d/a)^1 (fd*c/a)y1(x) ( alpha E alpha B alpha D alpha A ) ^ (-) 1 ( alpha F alpha D alpha C alpha A ) STO Y 1 ( X ) ENTER :DispG: Pause: ClrIO CATALOG D [arrow down to DispG] ENTER 2nd : CATALOG P [arrow down to Pause] ENTER 2nd : CATALOG C [arrow down to ClrIO] ENTER ENTER alpha M 2nd [ 2 , 2 2nd ] STO alpha G ENTER F[IfThen] 1 [IfThenEndIf] alpha G 2nd MATH 8 [Test] 6 [] 0 [arrow to the end of Then] ENTER

:m[2, 2] g :If g 0 Then

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:mRow (g^-1, m, 2 )m 2nd MATH 4 [Matrix] alpha J 3 [mRow(] alpha G ^ (-) 1 , alpha M , 2 ) STO alpha M ENTER :Disp OBTAIN LEADING F[Disp] 2nd 2nd ENTER :Disp 1 IN ROW 2 :Pause m a-lock O B T A I N alpha IN LEADING R O W alpha

F[Disp] 2nd alpha 2nd ENTER
CATALOG P [arrow down to Pause] ENTER alpha M ENTER
:ClrDraw: DispGraph: Pause: ClrIO CATALOG C [arrow down to ClrDraw] ENTER 2nd : CATALOG D [arrow down to DispG] ENTER 2nd : CATALOG P [arrow down to Pause] ENTER 2nd : CATALOG C [arrow down to ClrIO] ENTER ENTER :Disp OBTAIN 0 ABOVE F[Disp] 2nd a-lock O B T A I N lock A B O V E 2nd ENTER :Disp LEADING 1 IN :Disp COLUMN 2 :m[1, 2] h F[Disp] 2nd L E A D I N G I N 2nd ENTER F[Disp] 2nd C O L U M N alpha 1 alpha 0 a-lock a-

alpha 2 2nd ENTER

alpha M 2nd [ 1 , 2 2nd ] STO alpha H ENTER
:mRowAdd (h, m, 2, 1)m 2nd MATH 4 [Matrix] alpha J [Row ops] 4 [mRowAdd(] (-) alpha H , alpha M , 2 , 1 ) STO alpha M ENTER :Pause m: ClrDraw: FnOff 2 CATALOG P [arrow down to Pause] ENTER alpha M 2nd : CATALOG C [arrow down to ClrDraw] ENTER 2nd : CATALOG F [arrow down to FnOff] ENTER 2 ENTER :m[1, 3] j :LineVertical j :DispG: Pause: ClrIO alpha M 2nd [ 1 , 3 2nd ] STO alpha J ENTER CATALOG L [arrow down to LineVert] ENTER alpha J ENTER CATALOG D [arrow down to DispG] ENTER 2nd : CATALOG P [arrow down to Pause] ENTER 2nd : CATALOG C [arrow down to ClrIO] ENTER ENTER F[Disp] 2nd 2nd ENTER a-lock T H E POINT OF

:Disp THE POINT OF

:Disp INTERSECTION IS F[Disp] 2nd I N T E R S E C T I O N ENTER

I S 2nd

:Disp x = , m[1, 3], y = , m[2, 3] F[Disp] 2nd alpha X = 2nd , alpha M 2nd [ 1 , 3 2nd ] , 2nd Y = 2nd , alpha M 2nd [ 2 , 3 2nd ] ENTER :Stop :EndIf CATALOG S [arrow down to Stop] ENTER ENTER [arrow to the end of EndIf] ENTER

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:If m[2, 3] = 0 Then
F[IfThen] 1 [IfThenEndIf] alpha M 2nd [ 2 , 3 2nd ] = 0 [arrow to the end of Then] ENTER a-lock I N F I N I T E L Y MANY
:Disp INFINITELY MANY F[Disp] 2nd 2nd ENTER :Disp SOLUTIONS :Stop :Else :Disp INCONSISTENT :Disp SYSTEM :Stop :EndIf :EndPrgm
F[Disp] 2nd S O L U T I O N S 2nd ENTER CATALOG S [arrow down to Stop] ENTER ENTER CATALOG E [arrow down to Else] ENTER ENTER F[Disp] 2nd I N C O N S I S T E N T 2nd ENTER F[Disp] 2nd S Y S T E M 2nd ENTER CATALOG S [arrow down to Stop] ENTER ENTER [arrow to the end of EndIf] ENTER
When you have finished, press 2nd QUIT to leave the program editor. You may remove a program from memory by pressing 2nd VAR-LINK. Then move the cursor to the programs name and press ENTER to delete the entire program. III.4.2 Running the Program: To execute the program you have entered, go to the Home screen and type the name of the program, including the parentheses and then press ENTER. If you have forgotten its name, press 2nd VAR-LINK to list all the variables that exist. The programs will have PRGM after the name. You can execute the program from this screen by highlighting the name and then pressing ENTER. The screen will return to the Home screen and you will have to enter the closing parentheses ) and press ENTER to execute the program. The program has been written to prompt you for values of the coefficients a, b, c, d, e and f in two linear equations, ax by c and dx ey f. Input each value, then press ENTER to continue the program. This demonstration is most effective for equations that do not correspond to vertical or horizontal lines, and whose y-intercepts are between 10 and 10. While the demonstration is running, note that each elementary row operation creates an equivalent system. The equivalence is reinforced graphically by the fact that while a row operation may change the slope of the lines, their point of intersection remains constant. When the program comes to a pause, press ENTER to continue the program. If you need to interrupt the program during execution, press ON and then ENTER. The instruction manual for your TI-89 gives detailed information about programming. Refer to it to learn more about programming and how to use other features of your calculator.