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TI-92 - Cengage Learning

1. Copyright Houghton Mifflin Company All rights reserved 41 Graphing Technology Guide TI 92 lw ntnt cos x2 x 0 1 9045242379 E 80 arran UNIT Figure 5 102 Using nint 5 12 2 Areas You may approximate the area under the graph of a function y f x between x A and x B with your TI 92 To do this you use the F5 Math menu when you have a graph displayed For example here are the keystrokes for finding the area under the graph of the function y cos x between x 0 and x 1 The area is represented by the definite integral cos x dx First clear any existing graphs and then press COS X A 2 ENTER followed by GRAPH to draw the graph The range in Figure 5 103 extends from 5 to 5 horizontally and from 2 to 2 vertically Now press F5 Math 7 f x dx The TI 92 will prompt you for the lower and upper limits which are entered by pressing 0 ENTER 1 ENTER The region between the graph and the x axis from the lower limit to the upper limit is shaded and the approximate value of the integral is displayed Figure 5 104 Technology Tip If the function takes on negative values between the lower and upper limits the value that the TI 92 displays it the value of the integral not the area of the shaded region a Aid N bine PT A zoon trace A e Zo0n Trace ReGraphath prau SEC dx 90452424 Fa TAD PAE TE m cus TE Figure 5 103 Graph of y cos x Figure 5 104 Gra
2. TEE AEE THE Figure 5 90 Parametric graph of x cos and y sin t You may Zoom and Trace along parametric graphs just as you did with function graphs However unlike with function graphs the cursor will not move to values outside of the t range so will not work when t 0 and gt will not work when t 2x As you trace along this graph notice that the cursor moves in the counterclockwise direction as t increases 5 8 2 Rectangular Polar Coordinate 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 Of course you may override the current angle measure setting as explained in Section 5 4 1 For the following examples the TI 92 is set to radian measure Given the rectangular coordinates x y 4 3 convert to polar coordinates r 0 in the Home screen by pressing 2nd MATH 2 Angle 5 R Pr 4 3 ENTER The value of r is displayed now press 2nd MATH 2 Angle 6 R PO 4 3 ENTER to display the value of Figure 5 91 The polar coordinates are approximately 5 0 6435 Suppose r 3 x Convert to rectangular coordinates x y by pressing 2nd MATH 2 Angle 3 P gt Rx 3 2nd x ENTER The x coordinate is displayed Press 2nd MATH 2 Angle 4 P Ry 3 2nd x ENTER to display the
3. n b and the exponential growth model in this case is y 44516 or y 4451 0901700 iz ORAGARRE AEA STAT VARS ETletat yea bx a 4450 972591 b 1 017454 Enter 0K Figure 5 73 Exponential growth model If you wish to plot and graph the data follow the method for linear regression Set an appropriate range for the data and then press GRAPH The data will now be plotted in the range To graph the regression equation also store the regression equation to a y plot that is free As in the linear regression model press Y and inactivate or clear any other existing functions then press GRAPH to graph the exponential growth model Note that the exponential regression model does not need to be converted to the form y ae before graphing Remember to clear or deselect the plot before viewing graphs of other functions 5 6 Matrices 5 6 1 Making a Matrix The T1 92 can work with as many different matrices as the memory will hold Here s how 1 4 3 5 to create this 3x4 matrix 1 3 1 3 in your calculator 20 4 6 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 5 74 Press ENTER to accept these values Copyright Houghton Mifflin Company All rights reserved 27 Graphing Technology Guide TI 92
4. CXE Figure 5 67 Entering data points 5 5 2 Plotting Data First check the MODE screen Figure 5 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 Y and ENTER to select the Plot Type as Scatter and the Mark as a Box Use the keyboard to set the independent variable x to 1 and the dependent variable y to c2 as shown in Figure 5 68 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 5 69 shows this plot in a window from 0 to 10 horizontally and vertically You may now press F3 Trace to move from data point to data point To draw the scatter plot in a window adjusted automatically to include all the data you entered press F2 Zoom 9 ZoomData Copyright Houghton Mifflin Company All rights reserved 25 Graphing Technology Guide TI 92 ointorize Plott d Plot Type Scatter Py Mark Box gt I fer 2 3 al 5 HE 2 r ian Tea APPA Fane Figure 5 68 Plot1 menu Figure 5 69 Scatter plot When you no longer want to see the scatter plot press APPS 6 Data Matrix Ed
5. 2x 1 and y x 3x 4 correspond to the solutions of the single equation x 3x7 2x 1 x7 3x 4 which simplifies to x 2x x 3 0 So you may also graph x 2x x 3 and find its x intercepts to solve the system or use the solve function 5 3 4 Solving Inequalities by Graphing Consider the inequality 1 2 gt x 4 To solve it with your TI 92 graph 25 and y x 4 Figure 5 61 First locate their point of intersection atx 2 The the two functions y inequality is true when the graph of y 1 3 lies above the graph of y x 4 and that occurs when x lt 2 So the solution is the half line x lt 2 or 2 Copyright Houghton Mifflin Company All rights reserved 21 Graphing Technology Guide TI 92 fv amp zoon trace ReGraphlMath rauly A xot2 yer gt Figure 5 61 Solving iRexna aia Pee eoon rrace ii Ha Figure 5 63 Graph of y gt The TI 92 is capable of shading the region above or below a graph or between two graphs For example to graph y x 1 first enter the function y x 1 as y1 Then highlight y1 and press F6 Style 7 Above sce Figure 5 62 These keystrokes instruct the TI 92 to shade the region above y x 1 Press GRAPH to see the graph The region above the graph will be shaded using the first shading option of vertical lines as in Figure 5 63 Now use shading to solve the previous inequalit
6. Complex roots m n DISP 2nd tCOMPLEX ROOTS 2nd M NENTER displays the message Complex roots and both roots Stop t STOP ENTER Lbl Label2 LBL LABEL2ENTER Disp Double root m DISP 2 2nd tDOUBLE ROOT2nd M ENTER displays the message Double root and the solution root When you have finished press 2nd QUIT to leave the program editor and move on If you want to remove a program from memory press 2nd VAR LINK use the cursor pad 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 calculator s memory Technology Tip The program uses the variables a b c d m and n Note that any previous values for these variables including matrices will be replaced by the values used by the program The TI 92 does not distinguish between A and a in these uses Note that you will have to clear the variables using 2nd VAR LINK in order to use these names again in the current folder From the Home screen F6 will clear all 1 character variables Another way to deal with this is to create a new folder From the Home screen press F4 Other B NewFold and type the name of the new folder The work you do from that point on will be in the new folder as indicated by the folder name in the lower left corner of the Status line You can change folders from the MODE menu or from the Home screen by typing setFold foldername
7. where foldername is the existing folder that you wish to be in 5 10 2 Executing a 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 to execute it 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 retum to the Home screen and you will have to enter the closing parenthesis and press ENTER to execute the program The program has been written to prompt you for values of the coefficients a b and c in a quadratic equation ax bx c 0 Input a value then press ENTER to continue the program If you need to interrupt a program during execution press ON After the program has run the TI 92 will display the appropriate message and the root s The TI 92 will be on the Program I O screen not the Home screen The F5 key toggles between the Home screen and the Program I O screen or you can use 2nd QUIT HOME to go to the Home screen or the APPS menu to go any screen The instruction manual for your TI 92 gives detailed information about programming Refer to it to learn more about programming and how to use other features of your calculator Copyright Houghton Mifflin Company All rights reserved 38 Graphing Technology Guide TI 92 5 11 Differ
8. 3 35 17 this matrix as e press A STO 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 and replaces the previous matrix e The last two steps of the row operations are shown in Figure 5 80 Copyright Houghton Mifflin Company All rights reserved 29 Graphing Technology Guide TI 92 Row Operations Keystrokes add row T to row 2 2nd MATH 4 D 2 E 1 2 STOVE ENTER add 2 times row 1 to row 3 2nd MATH 4 D 4 2 E 1 3 STOVE ENTER add row 2 to row 3 2nd MATH 4 D 2 2 3 STOh E ENTER Itiply row 3 by 4 ie row 2 by 2nd MATH 4 D 3 1 2 E 3 STOWE ENTER E fisebra csie 3 ls rourdeke 2 3 e 5 4 s Je mRow 1 2 e 3 2 js de 5 EAEE ae peti ara Figure 5 80 Final matrix after row operations Thus 2 2 soy l andx 1 Technology Tip The TI 92 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 A ENTER and the reduced row echelon form is obtained by pressing 2nd MATH 4 Matrix 4 rref 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 92 produces will result in the same solution to the system 1 23 5 6 4 Determinants and Inv
9. a ae pce ATHE TRIE Figure 5 78 Swap rows 2 and 3 Figure 5 79 Add 4 times row 2 to row 3 To interchange the second and third rows of the matrix a that was defined above press 2nd MATH 4 Matrix D Row ops 1 rowSwap A 2 3 ENTER sce Figure 5 78 The format of this command is rowSwap matrix rindex1 rindex2 To add row 2 and row 3 and store the results in row 3 press 2nd MATH 4 Matrix D Row ops 2 rowAdd A 2 3 ENTER The format of this command is rowAdd matrix1 rindex1 rindex2 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 D Row ops 3 mRow 4 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 D Row ops 4 mRowAdd 4 A 2 3 ENTER see Figure 5 79 The format of this command is m RowAdd expression matrix1 Index1 Index2 Note that your TI 92 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 row operations to solve this system of linear equations x 3y 4 leis 17 123 9 First enter this augmented matrix as a in your TI 92 1 3 0 4 Then return to the Home screen and store 2
10. as many as ninety nine different functions for the TI 92 to use at one time If there is already a function y1 press or W as many times as necessary to move the cursor to y1 and then press CLEAR to delete whatever was there Then enter the expression 1975 10x by pressing these keys 1975 1 0 X ENTER Now press HOME Copyright Houghton Mifflin Company All rights reserved 8 Graphing Technology Guide TI 92 se lroomledit Hiilstsiel gt Chai debralceiclotherlprantolciear a z uico 2198 1 1865 2161 5 rare ron T Figure 5 11 Y screen Figure 5 12 Evaluating a function Assign the value 2230 to the variable x by these keystrokes 2230 STO X ENTER Then press the following keystrokes to evaluate y1 and find January s 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 5 12 You may also have the TI 92 make a table of values for the function Press TblSet to set up the table Figure 5 13 Move the blinking cursor down to the fourth line beside Independent then press gt 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 5 14 Press W to move to the next line and
11. continue to enter additional values for x The TI 92 automatically completes the table with the corresponding values of y1 Press 2nd QUIT to leave the TABLE screen Technology Tip The TI 92 requires multiplication to be expressed between variables so xxx does not mean X rather it is a new variable named xxx Thus you must use either x s between the x s or A for powers of x Of course expressed multiplication is not required between a constant and a variable See your TI 92 manual for more information about the allowed usage of implied multiplication v amp Setuplce Header Del Rowling Rou Re i se5_ 2161 5 Graph lt gt Table OFF gt Independent CTFAUTO Ta AEE TT Figure 5 13 Table Setup screen Figure 5 14 Table of values 5 2 2 Functions in a Graph Window Once you have entered a function in the Y screen of the TI 92 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 x 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 5 15 Now press GRAPH and the TI 92 changes to a window with the graph of y x 4x While the TI 92 is calculating coordinates for a p
12. 0 5 ENTER 234000 Technology Tip Note that if you had set the calculation mode to either AUTO or EXACT the last line of page 2 of the MODE menu the TI 92 would display 4 for 24 and 2 In 5 3 In 2 for In 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 Afremain or VWYYVYYVYYYY 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 Note that you can select a function or a sub menu from the current menu by pressing either Y 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 the letter A than to press W nine times to get the remain function Algebra Hpperbot i string OPPE OR USE Er e ENTERED AND ESEE Figure 5 8 MATH menu and Number sub menu The factorial of a non n
13. 01 je nDerivlk k 1 0 4 k 2 5 18 75000001 riam Tao MEEDE TEH Figure 5 97 Using nDeriv The TI 92 has a function nDeriv which is available in the Calculus sub menu of the MATH menu that will fens So to find a numerical approximation tof 2 5 when fx x and with Ax 0 001 go to the Home screen and press 2nd MATH A Calculus AfnDeriv X 3 X 2nd X 2 5 ENTER as shown in Figure 5 97 The format of this command is nDeriv expression variable Ax calculate the symmetric difference Copyright Houghton Mifflin Company All rights reserved 39 Graphing Technology Guide TI 92 where the optional argument Ax controls the accuracy of the approximation The added expression 2nd X 2 5 give the value of x at which the derivative is evaluated The is found on the keyboard above the K so press 2nd K to enter it If no value for Ax is provided the TI 92 automatically uses Ax 0 001 If no value for x is given the TI 92 will give the symmetric difference as a function of x The same derivative is also approximated in Figure 5 97 using Ax 0 0001 For most purposes Ax 0 001 gives a very good approximation to the derivative Note that in Figure 5 97 any letter can be used for the variable Ta FY Te 2 Perens ihan Zoon Trace ReGraphivath orau r S x 2 vy 32 a 3 oe 38 2 GO nDeriv yl GO x ia nDer ty 1 OO ee aE EE Tan Figure 5 98 Entering f x an
14. 9 Note that any letter could have been used for the variable This is the reason that you must indicate to the TI 92 that the variable being used is X ree laigebralcsiclother prsntolciear az Is solve 24 x3 36 x 17 0 x x 1 41449606 17 gt solve C24x ca Figure 5 59 solve function Technology Tip To solve an equation like 24X 17 36x you may first transform it into standard form 24x 36x 17 0 and proceed as above However the solve function does not require that the function be in standard form You may also graph the fwo functions y 24x 17 and y 36x then zoom and trace to locate their point of intersection 5 3 3 Solving Systems by Graphing The solutions to a system of equations correspond to the points of intersection of their graphs Figure 5 60 For example to solve the system y x 3x 2x 1 and y x 3x 4 first graph them together Then use zoom and trace or the intersection option in the F5 Math menu to locate their point of intersection approximately 2 17 7 25 ae OTA Ti Figure 5 60 Graph of y X 3x7 2x 1 and y x 3x 4 If you do not use the Intersection option you must judge whether the two current y coordinates are sufficiently close for x 2 17 or whether you should continue to zoom and trace to improve the approximation The solutions of the system of two equations y x 3x
15. CHAPTER 5 TEXAS INSTRUMENTS TI 92 5 1 Getting Started with the TI 92 In this book the key with the green diamond symbol inside a green border will be indicated by the key with the white arrow pointing up inside a white border the shift key will be indicated by and the key with the white arrow the backspace key pointing to the left will be indicated by Although the cursor pad allows for movements in eight directions we will mainly use the four directions of up down right and left These directions will be indicated by Y gt and respectively There are eight blue keys on the left side of the calculator labeled F1 through F8 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 5 1 1 Basics Press the ON key to begin using your TI 92 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 Note that the TI 92 has three ENTER keys and two 2nd keys which can be used interchangeably Check your TI 92 s settings by pressing MODE If necessary use the cursor pad to move the blinking cursor to a setting you wan
16. D m E Piot Setup cel Raj pa siz k Pm T FABRE Type Matrix Folder main Usriableia J 2 Row dimensionig Col dinension 4 Erk EED r2ci 1 am aE TC ra TESTA Figure 5 74 Data Matrix menu Figure 5 75 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 press 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 ieft of the screen shows the cursor s current location within the matrix The element in the second row and first column in Figure 5 75 is highlighted so the lower left of the window is r2c1 1 showing that element s 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 5 6 2 Matrix Math From the Home screen you can perform many calculations with matrices To see matrix a press A ENTER Figure 5 76 Perform the scalar multiplication 2 a pressing 2A ENTER The resulting matrix is displayed on the screen To create matrix b as 2a press 2 A STOP B ENTER Figure 5 77 or if you do this immediately after calculating 2a press only STO B ENTER The calculator
17. I 92 One method is by using the APPS menu Figure 5 3 which is accessed by pressing the blue APPS key on the right side of the calculator Thus to get to the Home screen you can press 2nd QUIT HOME or APPS ENTER Progran Editor Fest Editor Figure 5 3 APPS menu 5 1 2 Editing One advantage of the TI 92 is that you can use the cursor pad to scroll in order to see a long calculation For example type this sum Figure 5 4 14243 44 5 64 74 849 10 11 12 13 14 15 16 17 18 19 20 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 gt in the history area You can scroll through the entire calculation by using the cursor pad f or W to put the cursor on the appropriate line and then using gt or to move the cursor to the part of the calculation that you wish to see F T i her PrantO clear a z lother lpr iu ES E lalek ti 2 3 4 5 6 7 8 9 10 11 12 EEEIEE 17418419421 Hae EAD APP FUNC 30 Figure 5 4 Home screen Often we do not notice a mistake until we see how unreasonable an answer is The TI 92 permits you to redisplay an entire calculation edit it easily then execute the corrected calculation Suppose you had typed 12 34 56 as in Figure 5 5 but had not yet pressed ENTER when you r
18. RY This makes it easy to use the answer from one computation in another computation For example press 30 15 ENTER so that 45 is the last result displayed Then press 2nd ANS 9 ENTER and get 5 because 45 9 5 The answer locations are indexed by ans and the entry locations are indexed by entry where indicates the number of the entry answer The pairs are numbered with the most recent computation as 1 Hence the number of a pair changes with each successive computation that is entered The number of an entry or answer can be found by using the cursor pad M to scroll up to the entry or answer The number which is the same for both the entry and the answer is shown on the bottom of the screen To use an earlier answer or entry in a computation to calculate say 15 times answer 3 plus 75 press 15 x AN S 3 7 5 ENTER using the keyboard to type the letters A N and S With a function like division you press the after you enter an argument For such functions whenever you would start a new calculation with the previous answer followed by pressing the function key you may press just the function key So instead of 2nd ANS 9 in the previous example you could have pressed simply 9 to achieve the same result This technique also works for these functions x a 2nd x Here is a situation where this is especially useful Suppose a person makes 5 85 per hour and you are asked to calculate earnings for a day a week and
19. TER to accept the regression equation and close the STAT VARS screen To see both the data points and the regression line Figure 5 72 go to the Plot Setup screen and select Plot1 then press GRAPH to display the graph Pp eoonltracelretaphbistnorsae P aR arr re Figure 5 72 Linear regressi Copyright Houghton Mifflin Company All rights reserved 26 Graphing Technology Guide TI 92 5 5 4 Exponential Growth Model The table shows the world population in millions from 1980 to 1992 Year 1980 1985 1986 1987 1988 1989 1990 1991 1992 Population 4453 4850 4936 5024 5112 5202 5294 5384 5478 millions Clear the previous data by going to the current variable in the Data Matrix Editor and pressing F1 Tools 8 Clear Editor ENTER Follow the procedure described above to enter the data in order to find an exponential model that approximates the given data Use 0 for 1980 5 for 1985 and so on The TI 92 will not compute the exponential growth model y ae The exponential regression that the TI 92 will compute is of the form y ab To get this exponential growth model press F5 Calc and set the Calculation Type to 4 ExpReg the x variable to c1 and the y variable to c2 Then press ENTER to find the values of a and b Figure 5 73 In this case the exponential growth model is y 4451 1 017454 To convert this to the form y ae the required equation is
20. TI 92 will allow you to continue the trace by panning the viewing rectangle Check the WINDOW screen to see that the xmin and xmax are automatically updated The TI 92 has a display of 239 horizontal columns of pixels and 103 vertical rows so when you trace a curve across a graph window you are actually moving from xmin to xmax in 238 equal jumps each called Ax You would xmax xmin B8 numbers like 0 1 or 0 25 so that when you trace along the curve the x coordinates will be incremented by such a convenient amount Just set your viewing rectangle for a particular increment Ax by making xmax xmin 238 Ax For example if you want xmin 5 and Ax 0 3 set xmax 5 238 0 3 66 4 Likewise set ymax ymin 102 Ay if you want the vertical increment to be some special A calculate the size of each jump to be Ax Sometimes you may want the jumps to be friendly To center your window around a particular point say h k and also have a certain Ax set xmin h 119 Ax and make xmax h 119 Ax Likewise make ymin k 51 Ay and make ymax k 51 Ay 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 0 119 0 25 29 75 xmax 0 119 0 25 29 75 ymin 0 51 0 25 12 75 and ymax 0 51 0 25 12 75 See the benefit by first plotting y x 2x 1 in a standard graphing window Trace near its y intercept w
21. TORE Fam TA AFFET Tan 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 ToolBar menu Figure 5 23 and then pressing 9 Format to display the Format menu Figure 5 24 or by pressing F as indicated on the ToolBar menu in Figure 5 23 Use the cursor pad to move the blinking cursor to Grid press 2 On ENTER to redraw the graph Figure 5 25 shows the same graph as in Figure 5 22 but with the grid turned on Copyright Houghton Mifflin Company All rights reserved 11 Graphing Technology Guide TI 92 5A EA SN KAA ana iors Trace sis prasl 7 z in t ue Copy Fe Coordinates Graph Order Grid 2 Pre on Leading turgor or _ Cabelas 1 OFF SiPornat na EEND TEE E ETEN Figure 5 23 ToolBar menu Figure 5 24 Format menu Eorl aceledraenkatnlorsul 7 Sa ara ra Figure 5 25 Grid turned on for y x 4x In general you ll want the grid turned off so do that now by pressing F and turning the Grid option to OFF then pressing ENTER 5 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 92 the greatest integer function is called floor and is located under the N
22. a year Execute the given keystrokes to find the person s incomes during these periods results are shown in Figure 5 7 Pay Period Keystrokes Earnings 8 hour day 5 85 x 8 ENTER 46 80 S day week x 5 ENTER 234 52 week year x 52 ENTER 12 168 ee algebra 5 85 8 48 3 js 46 8 5 234 m 234 52 12168 aie a ar TRI Figure 5 7 ANS variable 5 1 7 The MATH Menu Operators and functions associated with a scientific calculator are available either immediately from the keys of the TI 92 or by the 2nd keys You have direct access to common arithmetic operations 2nd J 2nd x a trigonometric functions SIN COS TAN and their inverses 2nd SIN 2nd COS 2nd TAN exponential and logarithmic functions LN 2nd e and a famous constant 2nd x Copyright Houghton Mifflin Company All rights reserved 6 Graphing Technology Guide TI 92 A significant difference between the TI 92 graphing calculators and most scientific calculators is that TI 92 requires the argument of a function after the function as you would see in a formula written in your textbook For example on the TI 92 you calculate V16 by pressing the keys 2nd V 16 in that order Here are keystrokes for basic mathematical operations Try them for practice on your TI 92 Expression Keystrokes Display Bizy and J 342 4 1 2 ENTER 5 24 2 3 2nd x ENTER 2 333333333 In 200 LN 200 ENTER 5 298317367 2 34 10 2 34 x 1
23. able A Input Enter b b TINPUT o 2nd ENTER B 2nd B ENTER Input Enter c c TINPUT o 2nd t ENTER o C 2nd C ENTER b 4 arc gt d Ba2 4xAxCSTObD ENTER calculates the discriminant and stores its value as d Copyright Houghton Mifflin Company All rights reserved 36 Graphing Technology Guide TI 92 V d 2a gt m B 2nd D 2A STOP M ENTER calculates one root and stores it as m b V d 2a gt n B 2nd J D 2A STON ENTER calculates the other root and stores it as n lfd lt 0 Then TIF o D2nd lt 0 THEN ENTER tests to see if the discriminant is negative Goto Label1 t GOTO tLABEL1ENTER if the discriminant is negative jumps to the line Label I below Endif TEND TIFENTER if the discriminant is not negative continues on to the next line Ifd 0 Then TIF D 0 2 THEN ENTER tests to see if the discriminant is zero Goto Label2 t GOTO tTLABEL2ENTER if the discriminant is zero jumps to the line Label 2 below Endif TEND TIFENTER if the discriminant is not zero continues on to the next line Disp Two real roots m n DISP o 2nd t TWO REAL ROOTS 2nd M NENTER displays the message Two real roots and both roots Stop STOPENTER stops program execution Lbl Label1 LBL tLABEL1ENTER jumping point for the Goto command above Copyright Houghton Mifflin Company All rights reserved 37 Graphing Technology Guide TI 92 Disp
24. ametric Equations The TI 92 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 5 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 cos and y sin t First check that angle measure is in radians Then press Y COS T 3 ENTER SIN T 3 ENTER Figure 5 88 Press WINDOW to set the graphing window and to initialize the values oft In the standard window the values oft go from 0 to 2r in steps of BA 0 1309 with the view from 10 to 10 in both directions In 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 5 89 Now press GRAPH to draw the graph Figure 5 90 Ee Alietatel Eoo TF Gi 85318530718 a 13089389563957 t3973 TERES TE GIC TA EET TE Figure 5 88 x cos t and y sin t Figure 5 89 Parametric WINDOW menu Copyright Houghton Mifflin Company All rights reserved 33 Graphing Technology Guide TI 92 2600 IRENA RGA FaceReGraphMathbraule A _ EI
25. as b which can be edited Return to the Home screen HOME and press A 2nd x x B ENTER to get the answer as shown in Figure 5 83 TAA AAS ig eirda oter redea az ato Mam ETEA TRI Figure 5 83 Solution matrix The solution is still x 1 y l and z 5 7 Sequences 5 7 1 Iteration with the ANS key The ANS key enables you to perform iteration the process of evaluating a n function repeatedly As an example calculate z 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 92 calculator See the results in Figure 5 84 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 Display 1 27 ENTER 27 2 2nd ANS 1 3 ENTER 8 666666667 3 ENTER 55555556 4 ENTER 5185185185 Copyright Houghton Mifflin Company All rights reserved 31 Graphing Technology Guide TI 92 NAAR 7 P aigebralcaiclother prentolciear a z z at 8 666666667 2 6666666666667 1 A oai 2 5555555555556 1 PFPA nT ATTAT Figure 5 84 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 5 7 2 Arithmetic and Geometric Sequences Use iteration with the ANS variable to determi
26. as necessary You can also find the point of intersection of two graphs by pressing F5 Math 5 Intersection Trace with the cursor first along one graph near the intersection and press ENTER then trace with the cursor along the other graph and press ENTER Marks are placed on the graphs at these points Then set lower and upper bounds for the x coordinate of the intersection point and press ENTER again Coordinates of the intersection will be displayed at the bottom of the window Figure 5 58 Perheet era Figure 5 58 An intersection of y x 4x and y 25x 5 3 2 Solving Equations by Graphing Suppose you need to solve the equation 24x 36x 17 0 First graph y 24x 36x 17 in a window large enough to exhibit all its x intercepts corresponding to all the equation s zeros roots Then use trace and zoom or the TI 92 s zero finder to locate each one In fact this equation has just one solution approximately x 1 414 Copyright Houghton Mifflin Company All rights reserved 20 Graphing Technology Guide TI 92 Remember that when an equation has more than one x intercept it may be necessary to change the viewing rectangle a few times to locate all of them The TI 92 has a solve function To use this function you must be in the Home screen To use the solve function press SOLVE 24 X a 3 35 X 17 0 X ENTER The TI 92 displays the value of the zero Figure 5 5
27. ayed 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 92 s Home screen can be entered as a line in a program There are also special programming commands Input the program Quadrat by pressing the keystrokes given in the listing below 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 the Variable list highlight this program s name and press ENTER twice Each time you press ENTER while writing a program the TI 92 automatically inserts the character at the beginning of the next line The instruction manual for your TI 92 gives detailed information about programming Refer to it to learn more about programming and how to use other features of your calculator Note that this program makes use of the TI 92 s ability to compute complex numbers Make sure that Complex Format on the MODE screen Figure 5 1 is set to RECTANGULAR Enter the program Quadrat by pressing the given keystrokes A space entered by using the spacebar on the keyboard is indicated by Program Line Keystrokes Input Enter a a INPUT o 2nd tENTER A 2nd A ENTER displays the words ENTER A on the TI 92 screen and waits for you to input a value that will be assigned to the vari
28. cursor as close as you can to the point of intersection near x 2 see Figure 5 47 Then press ENTER and the calculator draws a magnified graph centered at the cursor s position Figure 5 48 The range variables are changed to reflect this new viewing rectangle Look in the WINDOW menu to verify this TOS Re Cee ebonlacelRedraphirath orauly 7 ireket New Center xc 1 6806723 9803922 6806723 yc 9803922 za rE Ne ae aE TRE Figure 5 47 Before a zoom in Figure 5 48 After a zoom in As you see in the F2 Zoom menu Figure 5 43 the TI 92 can zoom in press F2 Zoom 2 or zoom out press F2 Zoom 3 Zoom out to see a larger view of the graph centered at the cursor position You can change the horizontal and vertical scale of the magnification by pressing F2 Zoom C SetFactors see Figure 5 49 and editing xFact and yFact the horizontal and vertical magnification factors The zFact is only used when dealing with 3 dimensional graphs Copyright Houghton Mifflin Company All rights reserved 17 Graphing Technology Guide TI 92 PFE lzecn irace lees apnat oe oR zFactt Enter Ok O CESC CANCEL ae Ta wr Fi Figure 5 49 ZOOM FACTORS menu Technology Tip An advantage of zooming in from square viewing window is that subsequent windows will also be square Likewise if you zoom in from a friendly viewing rectangle the zoomed windows will also be friendly The defaul
29. d can help the TI 92 work more efficiently and quickly ET boonltracele Nera ra con aie Figure 5 52 Relative minimum on y x 4x Note that if you have more than one graph on the screen the upper right corner of the TI 83 screen will show the number of the function whose minimum maximum is being calculated 5 3 Solving Equations and Inequalities 5 3 1 Intercepts and Intersections Tracing and zooming are also used to locate an x intercept of a graph where a curve crosses the x axis For example the graph of y x amp x crosses the x axis three times Figure 5 53 After tracing over to the x intercept point that is farthest to the left zoom in Figure 5 54 Continue this process until you have located all three intercepts with as much accuracy as you need The three x intercepts of y x 8x are approximately 2 828 0 and 2 828 Ce eeinlrracekeGraphhistnbraule A am a PA TE Ean PE TT Figure 5 53 Graph of y x 8x Figure 5 54 Near an x intercept of y x 8x Technology Tip As you zoom in you may also wish to change the spacing between tick marks on the x axis so that the viewing rectangle shows scale marks near the intercept point Then the accuracy of your approximation will be such that the error is less than the distance between two tick marks Change the x scale on the TI 92 from the WINDOW menu Move the cursor down to xscl and enter an app
30. d f x Figure 5 99 Graphs of f x and f x Technology Tip It is sometimes helpful to plot both a function and its derivative together In Figure 5 99 the 5 x A pis function f x 7 and its numerical derivative actually an approximation to the derivative given by the ve symmetric difference are graphed on viewing window that extends from 6 to 6 vertically and horizontally You 5x 2 Tl 2nd MATH A Calculus A nDeriv Y 1 X X ENTER Figure 5 98 can duplicate this graph by first entering for y1 and then entering its numerical derivative for y2 by pressing Graphing the derivative will be quite slow Making the xres value larger on the WINDOW screen will speed up the plotting ofthe graph Technology Tip To approximate the second derivative f x of a function y f x or to plot the second derivative first enter the expression for y1 and its derivative for y2 as above Then enter the second derivative for y3 by pressing 2nd MATH A Calculus A nDeriv Y 2 X X ENTER You may also approximate a derivative while you are examining the graph of a function When you are in a graph window press F5 Math 6 Derivatives 1 dy dx then use the cursor pad to trace along the curve to a point where you want the derivative or enter a value and press ENTER For example with the TI 92 in Function graphing 5x 2 mode graph the function f x E in the standard viewing rectangle Then press F5 Math 6 Der
31. d 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 keyboard The format of the when function is when condition trueResult falseResult unknownResult where the falseResult and unknownResult are optional arguments 2 lt 0 For example to graph the function f x L x20 you want to graph x 2 when the condition x lt 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 W H E N X 2nd lt 0 X 2 1 X 1 ENTER Figure 5 28 Then press GRAPH to display the graph Figure 5 29 shows this graph in a viewing rectangle from 5 to 5 in both directions This was done in Dot style since the TI 92 will incorrectly connect the two sides of the graph at x 0 if the function is graphed in Line style cu En loonlesit alilstaiels lt s eketa 2i o zuhen x0 x212 x1 Ea Regret Tae a TPE TT Figure 5 28 Piecewise defined function Figure 5 29 Piecewise defined graph Other test functions such as lt gt and as well as logic operators can be found on the Test sub menu of the 2nd MATH menu 5 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 w
32. e period and amplitude to establish better WINDOW values Technology Tip Since n 27 1 when in radian mode set xmin 0 and xmax 6 2 to cover the interval from 0 to Next graph y tan x in the standard window first then press F2 Zoom 7 Zoom Trig to change to a special window for trigonometric functions in which the horizontal increment is 4 or 7 5 and the vertical range is from 4 to 4 The TI 92 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 5 2 3 when you plot the tangent function 5 5 Scatter Plots 5 5 1 Entering Data The table shows the total prize money in millions of dollars awarded at the Indianapolis 500 race from 1981 to 1989 Source Indianapolis Motor Speedway Hall of Fame Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 Prize S161 207 241 280 327 400 449 503 5 72 Smillion We ll now use the TI 92 to construct a scatter plot that represents these points and find a linear model that approximates the given data The TI 92 holds data in lis s You can create as many list names as your TI 92 memory has space to store Before entering data clear the data in the lists that you want to use To delete a list press 2nd VAR LINK This will display the list of folders showing the variables defined in each fo
33. ealize that 34 should have been 74 Simply press the direction on the cursor pad 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 right of the 4 and press to delete the 4 Copyright Houghton Mifflin Company All rights reserved 3 Graphing Technology Guide TI 92 ete De Sea e Se Ger rarorioriiti2y 210 TE TRT Figure 5 5 Editing a calculation Technology Tip The TI 92 has two different inputing modes insert and overtype The default mode is the insert mode in which the cursor is a blinking vertical line and new text will be inserted at the cursor s 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 92 remains in whatever the last input mode was even after being turned off 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 the direction on the cursor pad moves the cursor to the beginning of the line while pressing the gt direction on the cursor pad puts the c
34. ecomes 5000 1 085 Here are the keystrokes for finding the balance after 3 5 and 10 years Years Keystrokes Balance 3 5000 1 085 3 ENTER 36 386 45 gt 5ENTER 7 518 28 10 gt 10 ENTER 11 304 92 CPRaisebraksiclotherlprantolciear a z 5000 1 085 gt 6386 445625 s000 1 085 5 7518 283451 5000 1 085 1 11304 91721 SO00 lt 1 085 gt 10 aie RAT A ROT Figure 5 6 Editing expressions Copyright Houghton Mifflin Company All rights reserved 4 Graphing Technology Guide TI 92 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 gt 5 7 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 gt CLEAR 5 7 ENTER 5 1 3 Key Functions Most keys on the TI 92 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
35. egative 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 1 2 3 4 24 You will learn more about applications of factorials in your textbook but for now use the TI 92 to calculate 4 Press these keystrokes 4 2nd MATH 7 Probability 1 ENTER On the TI 92 it is possible to do calculations with complex numbers To enter the imaginary number i press 2nd i For example to divide 2 3i by 4 2i press 2 3 2nd 4 2 2nd i ENTER The result is 0 1 0 8 Figure 5 9 To find the complex conjugate of 4 5i press 2nd MATH 5 Complex ENTER 4 5 2nd i ENTER Figure 5 9 Copyright Houghton Mifflin Company All rights reserved 7 Graphing Technology Guide TI 92 PP laiaebralcaiclother Prantolciear a z HH 14 8 ls conjc4 5 4 4 5 il ate con TREES Figure 5 9 Complex number calculations The TI 92 can also solve for the real and complex roots of an equation This is done by using the Solve function which is not on any of the keys but can be found in the CATALOG From the Home screen pressing 2nd CATALOG gives an alphabetical list of all functions and operations available on the TI 92 You can scroll through the CATALOG page by page by pressing 2nd Y or if you know what letter the function starts with pressing the letter moves the cursor to the beginning of the listings for that l
36. entiation sin 4 sin 4x EMY Plot the graph of f x EE in a convenient x 5 11 1 Limits Suppose you need to find this limit lim PR g viewing rectangle that contains the point where the function appears to intersect the line x 0 because you want the limit as x 0 Your graph should lend support to the conclusion that lim 4 4 Figure 5 95 ars To test whether the conclusion that lim 2 is reasonable evaluate the function f x 2 for several large e eF x positive values of x since you want the limit as x os For example evaluate f 100 1000 and 10 000 2x l Another way to test the reasonableness of this result is to examine the graph of f x Z in a viewing rectangle xt that extends over large values of x See as in Figure 5 96 where the viewing rectangle extends horizontally from 0 2x to 90 whether the graph is asymptotic to the horizontal line y 2 Enter for y1 and 2 for y2 x l AA AA SN iA A 20on Trace ReGraph tath or AATA aes ta HERE TaRE a aE m Figure 5 95 Checking lim PE 4 Figure 5 96 Checking tim 2 x txt 5 11 2 Numerical Derivatives The derivative of a function fat x can be defined as the limit of the slopes of secant lines so f x lim A ie And for small values of Ax the expression funy fest Jim ve a good approximation to the limit gives ls nderiv x5 x x 2 5 18 7500
37. erses Enter this 3x3 square matrix as a 1 3 0 Since this consists of the first 2 5 5 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 in To calculate its L 23 determinant 1 3 0 go to the Home screen and press 2nd MATH 4 Matrix 2 det A ENTER You should 2 5 5 find that the determinant is 2 as shown in Figure 5 81 OPE i i er a2 Pee a ans deta 2 det lt jii TT TRA ne RCT Figure 5 81 Determinant of a Figure 5 82 Inverse of a Copyright Houghton Mifflin Company All rights reserved 30 Graphing Technology Guide TI 92 Since the determinant of the matrix is not zero it has an inverse matrix Press A 2nd x ENTER to calculate the inverse The result is shown in Figure 5 82 x 2y 3z 9 Now let s solve a system of linear equations by matrix inversion Once again consider x 3y 4 The 2x 5y 5z 17 123 coefficient matrix for this system is the matrix 1 3 0 which was entered as matrix a in the previous 2 5 5 9 example Now enter the matrix 4 as b Since b was used before when we stored 2a as b press APPS 17 6 Data Matrix Editor 2 Open gt 2 Matrix YY gt and use V to move the cursor to b then press ENTER twice to go to the matrix previously saved
38. etter The format of cSolve is CSolve expression variable For example to find the zeros of f x xX 4 14x 20 from the Home screen press 2nd CATALOG and move the cursor down to cSolve then press ENTER The display will return to the Home screen with cSolve on the entry line To complete the computation press X A 3 4X24 14X 20 0 X ENTER The TI 92 will display the real and complex roots of the equation as shown in Figure 5 10 ee wlargebralcsiclother Prentolciear a z ls cSolvelx3 4 x2 14 x 20 0 x x 1 3 i on xel 3 i on x icSolve ae TRAO A 4x20 Tune 1090 Figure 5 10 cSolve function All functions and commands found in the CATALOG can also be used by merely typing the command using the keyboard Hence in the Home screen you could also press C S O L V E X3 4XA2 14X 20 0 X ENTER to find the zeros of f x x 4x 14x 20 5 2 Functions and Graphs 5 2 1 Evaluating Functions Suppose you receive a monthly salary of 1975 plus a commission of 10 of sales Let x your sales in dollars then your wages W in dollars are given by the equation W 1975 10x If your January sales were 2230 and your February sales were 1865 what was your income during those months Here s one method to use your TI 92 to perform this task Press the Y key above the letter W or APPS 2 Y Editor to display the function editing screen Figure 5 11 You may enter
39. 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 V25 press 2nd Y 25 ENTER Technology Tip The TI 92 automatically places a left parenthesis after many functions and operators including LN 2nd e SIN COS TAN and 2nd J Ifa right parenthesis is not entered the TI 92 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 fine and then press the key For example if you are in EXACT calculation mode and want to find the approximate value of 4 45 press 2nd V 45 The QWERTY keyboard on the TI 92 is similar to a typewriter and can produce both upper and lower case letters To switch from one case to another press 2nd CAPS For a single upper case letter use the tt key There are also additional symbols available from the keyboard by using the 2nd and keys Some of the most commonly used symbols are marked on the keyboard but most are not See your TI 92 user s manual for more information 5 1 4 Order of Operations The TI 92 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 additio
40. he cursor s location The first line of the Graph Format menu Figure 5 24 has options for displaying the cursor s position in rectangular RECT or polar POLAR form Pe bonltracelnedraenitathloraue A T Bc 5235987 27320508 vert PE ns Figure 5 93 Polar graph of r 4 sin 0 5 9 Probability 5 9 1 Random Numbers The command rand generates numbers You will find this command in the Probability sub menu of the MATH menu in the Home screen Press 2nd MATH 7 Probability 4 rand ENTER to generate a random number between 0 and 1 Press ENTER to generate another number keep pressing ENTER to generate more of them If you need a random number between say 0 and 10 then press 10 2nd MATH 7 Probability 4 rand ENTER To get a random number between 5 and 15 press 5 10 2nd MATH 7 Probability 4 rand ENTER If you need the random number to be an integer between 1 and 10 inclusive press 2nd MATH 7 Probability 4 rand 10 ENTER For a random negative integer between 1 and 10 inclusive press 2nd MATH 7 Probability 4 rand 10 ENTER 5 9 2 Permutations and Combinations To calculate the number of permutations of 12 objects taken 7 at a time 12P7 press 2nd MATH 7 Probability 2 nPr 12 7 ENTER Figure 5 94 Thus 12P7 3 991 680 3991680 792 Figure 5 94 P and 12C For the number of combinations of 12 objects taken 7 at a time 12C7 press 2nd MATH 7 Probab
41. hich is 0 1 and move towards its x intercept which is 1 0 Then press F2 Zoom 4 ZoomDec and trace again near the intercepts 5 2 6 Zoom Plot again the two graphs for y x 4x and y 25x There appears to be an intersection near x 2 The TI 92 provides several ways to enlarge the view around this point You can change the viewing rectangle directly by pressing WINDOW and editing the values of xmin xmax ymin and ymax Figure 5 42 shows a new viewing rectangle for the range displayed in Figure 5 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 F TAS EE TE Figure 5 41 New WINDOW Figure 5 42 Closer view A more efficient method for enlarging the view is to draw a new viewing rectangle with the cursor Start again with a graph of the two functions y 4x and y 2Sx in a standard viewing rectangle Press F2 Zoom 6 ZoomStd for the standard viewing window Now imagine a small rectangular box around the intersection point near x 2 Press F2 Zoom 1 ZoomBox Figure 5 43 to draw a box to define this new viewing rectangle 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 rectangle you imagine Press ENTER to fix the corner where you moved the cursor it changes shape and becomes a blin
42. il two successive approximations differ by less than some predetermined value say 0 0001 Note that each time you press ENTER the TI 92 will use the current value of x and that value is changing as you continue the iteration ree hisebralcsiclotherPrentolciear a z le 1 9x chs Teeter a WDeriv yiGo x revna sx ID s 3 x WMerivuioo gt ehoar aaa ma ONT Tune 3 50 Figure 5 101 Newton s method Technology Tip Newton s Method is sensitive to your initial value for x so look carefully at the function s graph to make a good first estimate Also remember that the method sometimes fails to converge You may want to write a short program for Newton s Method See your calculator s manual for further information 5 12 Integration 5 12 1 Approximating Definite Integrals The TI 92 has a function nint which is available in the Calculus sub menu of the MATH menu that will approximate a definite integral For example to find a numerical approximation to J cosa dx go to the Home sereen and press 2nd MATH A Calculus Bfinint COS X 2 X 0 1 ENTER Figure 5 102 The format of this command is nint expression variable lower limit upper limit The algorithm that the TI 92 uses to calculate the numerical integral is adaptive and has an accuracy goal of six significant digits If it seems that this goal has not been achieved the calculator will display the warning Questionable accuracy
43. ility 3 nCr 12 7 ENTER Figure 5 94 Thus C 792 5 9 3 Probability of Winning A state lottery is configured so that each player chooses six different numbers from 1 to 40 If these six numbers match the six numbers drawn by the State Lottery Commission the player wins the top prize There are Cs ways for the six numbers to be drawn If you purchase a single lottery ticket your probability of winning is 1 in Cs Press 1 2nd MATH 7 Probability 3 nCr 40 6 ENTER to calculate your chances but don t be disappointed Copyright Houghton Mifflin Company All rights reserved 35 Graphing Technology Guide TI 92 5 10 Programming 5 10 1 Entering a Program The TI 92 is a programmable calculator that can store sequences of commands for later replay Here s an example to show you how to enter a usefull program that solves quadratic equations by the quadratic formula Press APPS 7 Program Editor to access the programming menu The TI 92 has space for many programs each named by a name you give it To create a new program now start by pressing 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 Quadrat and press ENTER twice to go to the program editor The program name and the beginning and ending commands of the program are automatically displ
44. ish to graph the circle of radius 6 whose equation is x 3 36 First solve for y and get an equation for the top semicircle y Y36 x and for the bottom semicircle y V36 x Then graph the two semicircles simultaneously Use the following keystrokes to draw this circle s graph First clear any existing functions on the Y screen Enter V36 x as y1 and J36 x as y2 see Figure 5 30 by pressing 2nd J 36 X a 2 ENTER 2nd J 36 X 2 ENTER Then press GRAPH to draw them both Figure 5 31 EG Ge eM zoomledie fiiltsie lt i SANT DAE Sire uge s6 x 8 EUR Ti Fam TAS aE T Figure 5 30 Two semicircles Figure 5 31 Circle s graph standard WINDOW Instead of entering 36 x as y2 you could have entered y1 as y2 and saved some keystrokes On the TI 92 try this by going into the Y screen and pressing f to move the cursor up to y2 Then press CLEAR Y 1 X ENTER Figure 5 32 The graph should be as before Copyright Houghton Mifflin Company All rights reserved 13 Graphing Technology Guide TI 92 x yi xD ECE Tab MEET TE Figure 5 32 Using y1 in y2 If your range were set to a viewing rectangle extending from 10 to 10 in both directions your graph would look like Figure 5 31 Now this does not look a circle because the units along the axes are not the same You need what is called a squa
45. itor 1 Current F2 Plot Setup highlight Plot 1 and use F4 v to deselect plot 1 The TI 92 still retains all the data you entered 5 5 3 Regression Line The TI 92 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 92 graph the regression equation set Store RegEQ to as y1 x as shown in Figure 5 70 Press ENTER and the TI 92 will calculate a linear regression model with the slope named a and the y intercept named b Figure 5 71 The correlation coefficient measures the goodness of fit of the linear regression 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 92 displays both the correlation coefficient and the coefficient of determination R na EESTE F D Le A i taicuiation Type LinReg gt rE Pi GAA jlstat x al i k2 2 Store RegEa to Y10097 K Use Freq and Categories NO a Fee l a Fae 7 Ean rar Tar Figure 5 70 Linear regression Calculate dialog box Figure 5 71 Linear regression model Press EN
46. ivatives 1 dy dx The coordinates of the point in the center of the range will appear To find the numerical derivative at x 2 3 press 2 3 ENTER Figure 5 100 shows the derivative at that point to be about 0 7746922 E E zoon trace keGraphnachloraul So dysdx 7746922 Ls ee 5 x xl Figure 5 100 Derivative of f x atx 23 Copyright Houghton Mifflin Company All rights reserved 40 Graphing Technology Guide TI 92 If more than one function is graphed you can use f and W to scroll between the functions Note that different options are available from pressing F5 Math 6 Derivatives depending on whether the function s being graphed are in FUNCTION PARAMETER or POLAR mode 5 11 3 Newton s Method With the T1 92 you may iterate using Newton s method to find the zeros of a function f f o Recall that Newton s Method determines each successive approximation by the formula x x As an example of the technique consider flx 2x x x 1 Enter this function as y1 and graph it in the standard viewing window A look at its graph suggests that it has a zero near x 1 so start the iteration by going to the Home screen and storing 1 as x Then press these keystrokes X Y 1 X 2nd MATH A Calculus A nDeriv Y1 X X STOb X ENTER ENTER Figure 5 101 to calculate the first two iterations of Newton s method Press ENTER repeatedly unt
47. king square Figure 5 44 Use the arrow keys again to move the cursor to the diagonally opposite corner of the new rectangle Copyright Houghton Mifflin Company All rights reserved 16 Graphing Technology Guide TI 92 Figure 5 45 Note that you can use the diagonal directions on the cursor pad 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 5 46 a oem hianost krcko senhara 7 oom ut oomDec RA oon tA oon Trig oon Int toomata lowrvmyausce ienory gt Kori 2nd Corner att BERON Aap Farber ise ct 1 372549 EEE EAE TEREN ase ESE ENTERED ARO ESTEN ET Figure 5 43 F2 Zoom menu Figure 5 44 One corner selected You may cancel the zoom any time before you press this last ENTER Press F2 Zoom once more and start over Press ESC or GRAPH to cancel the zoom or press 2nd QUIT to cancel the zoom and return to the Home screen Eeke esr erele errieta 2 Fare ates 78431373 FAIN LIE TiN LAIN FAD Avra Fut Figure 5 45 Box drawn Figure 5 46 New viewing rectangle You can also quickly magnify a graph around the cursor s location Return once more to the standard window for the graph of the two functions y x 4x and y 25x Press F2 Zoom 2 ZoomIn and then use the cursor pad to move the
48. lder Highlight the name of the list that you wish to delete Copyright Houghton Mifflin Company All rights reserved 24 Graphing Technology Guide TI 92 and press F1 Manage 1 Delete ENTER The TI 92 will ask you to confirm the deletion by pressing ENTER once more Now press APPS 6 Data Matrix Editor 3 New W P R Z E ENTER to open a new variable called PRIZE Figure 5 66 Press ENTER to then begin entering the variable values with the years going in column c1 Instead of entering the full year 198x enter only x Here are the keystrokes for the first three years 1 ENTER 2 ENTER 3 ENTER and so on then press gt to move to the next list Use the cursor pad to move up to the first row and press 1 61 ENTER 2 07 ENTER 2 41 and so on see Figure 5 67 EERE Figure 5 66 Entering a new variable 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 Thus 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 FE Util 1 Insert 1 cell and a new empty cell is open T i ES Prot sotue esi Mester cai lutii et at ae Ara 41 FN 5 27 a
49. lot it displays a 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 gt 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 gt and Copyright Houghton Mifflin Company All rights reserved 9 Graphing Technology Guide TI 92 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 92 s screen will now be divided either horizontally or vertically as you choose Figure 5 15 shows the graph and the Y screen with the settings shown in Figure 5 16 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 acce
50. ne the n th 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 For a geometric sequence whose common ratio is 4 start the progression with 2nd ANS x 4 ENTER Era LESAN AE SAN EA 7 re aigebralcaiclother prsntolcieara z SE xT SAFE ECET Figure 5 85 Sequential Y menu Figure 5 86 Sequence mode You can also define the sequence recursively with the TI 92 by selecting Sequence in the Graph type on the first page of the MODE menu see Figure 5 1 Once again let s find the 18th term of an arithmetic sequence whose first term is 7 and whose common difference is 4 Press MODE gt 4 Sequence ENTER Then press Y to edit any of the TI 92 s sequences u1 through 199 Make w1 n ul n 1 4 and w1 1 7 by pressing U 1 N 1 4 ENTER 7 ENTER Figure 5 85 Press 2nd QUIT to return to the Home screen To find the 18th term of this sequence calculate v1 18 by pressing U 1 18 ENTER Figure 5 86 Of course you could also use the explicit formula for the n th term of an arithmetic sequence f a n 1 d First enter values for the variables a d and n then evaluate the formula by pressing A N 1 D ENTER For a geometric sequence
51. ns and subtractions Enter these expressions to practice using your TI 92 Expression Keystrokes Display 7 53 7 5x 3 ENTER 8 7 5 3 7 5 3 ENTER 6 120 10 120 10 2 ENTER 20 120 10 120 10 2 ENTER 12100 24 24 2 3 ENTER 3 z aay 24 2 3 ENTER 1728 3 f e ERE tak 7 5 x 3 ENTER 36 NN 1 5 1 5 Algebraic Expressions and Memory Your calculator can evaluate expressions such as afier you have entered a value for N Suppose you want N 200 Press 200 STOP N ENTER to store the value 200 in memory location N Whenever you use N in an expression the calculator will substitute the value 200 until you make a change by storing another number in N Next enter the expression xD by typing Nx N 1 2 Copyright Houghton Mifflin Company All rights reserved 5 Graphing Technology Guide TI 92 N N 1 ENTER For N 200 you will find that 20100 Note that there is no distinction made between upper and lower case letters in this case The contents of any memory location may be revealed by typing just its letter name and then ENTER And the TI 92 retains memorized values even when it is tuned off so long as its batteries are good 5 1 6 Repeated Operations with ANS As many entry answer pairs as the history area shows are stored in memory The last result displayed can be entered on the entry line by pressing 2nd ANS while the last entry computed is entered on the entry line by pressing 2nd ENT
52. ph and area Technology Tip Suppose that you want to find the area between two functions y f x and y g x from x A and x B If f x gt g x for A lt x B then enter the expression f x g x and use the method above to find the required area Copyright Houghton Mifflin Company All rights reserved
53. ppear at the same point you left it xe 1 092437 yc 4 5098039 Faun AEE fin Figure 5 35 Free moving cursor Copyright Houghton Mifflin Company All rights reserved 14 Graphing Technology Guide TI 92 Press F3 TRACE to enable the left and right gt directions to move the cursor along 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 function s graph so the y coordinate is the calculated value of the function at the corresponding x coordinate Cee eonltraceleedraeninsthboraule A T xct 2 436975 yct4 7249191 ane TNE TPE EON aE Figure 5 36 TRACE Figure 5 37 Two functions Cee eonliraceledraphinathlorsule 7 a T wE Figure 5 38 y x 4x and y 25x Now plot a second function y 25x along with y x 4x Press Y and enter 25x for y2 then press GRAPH to sce both functions Figure 5 38 Notice that in Figure 5 37 there are check marks to the left of both y1 and y2 This means that both functions will be graphed In the Y screen move the cursor onto y1 and press F4 The check mark left of y1 should disappear Figure 5 39 Now press GRAPH and see that only y2 is plotted Figure 5 40 T mm TAD AFTES T Figure 5 39 only y2 acti
54. re viewing rectangle Press F2 Zoom 5 ZoomSqr and 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 2 times xmax xmin For example see the WINDOW in Figure 5 33 to get the corresponding graph in Figure 5 34 This method works because the dimensions of the TI 92 s display are such that the ratio of vertical to horizontal is approximately 3 ipa peor Pe leoon ltracelReGraphhath orauly A Fa EE FRE a ARE TRE i Fama 32 7 Figure 5 3 Figure 5 34 A square circle The two semicircles in Figure 5 34 do not meet because of an idiosynerasy in the way the TI 92 plots a graph 5 2 5 TRACE Graph the function y x 4x from Section 5 2 2 using the standard viewing rectangle Remember to clear any other functions in the Y screen Press any of the cursor directions WV and sce the cursor move from the center of the viewing rectangle The coordinates of the cursor s location are displayed at the bottom of the screen as in Figure 5 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 pad again and the free moving cursor will rea
55. ropriate value The x intercept ofa function s graph is a zero of the function so while viewing the graph press F5 Math Figure 5 50 and choose 2 Zero to find a zero of this function Set a lower bound and upper bound as described in Section 5 2 7 The TI 92 shows the coordinates of the point and indicates that it is a zero Figure 5 55 Copyright Houghton Mifflin Company All rights reserved 19 Graphing Technology Guide TI 92 Figure 5 55 A zero of y x 8x TRACE and ZOOM are especially important for locating the intersection points of two graphs say the graphs of y a 4x and y 25x Trace along one of the graphs until you arrive close to an intersection point Then press or W to jump to the other graph Notice that the x coordinate does not change but the y coordinate is likely to be different Figures 5 56 and 5 57 INath orsule TOS E TE Tyr gt Z00n Trace ReGraph zoon Trace ReGraph wath orau xc 2 04 yet 329664 Fm T reo Ta Figure 5 56 Trace on y x 4x Figure 5 57 Trace on y 25x When the two y coordinates are as close as they can get you have come as close as you now can to the point of intersection So zoom in around the intersection point then trace again until the two y coordinates are as close as possible Continue this process until you have located the point of intersection with as much accuracy
56. set the spacing between tick marks on the axes xres sets pixel resolution 1 through 10 for function graphs Copyright Houghton Mifflin Company All rights reserved 10 Graphing Technology Guide TI 92 Fa TEE AET Fan Figure 5 18 Standard WINDOW Technology Tip Small xres values improve graph resolution but may cause the TI 92 to draw graphs more slowly Use f and V to move up and down from one line to another in this list pressing the ENTER key will move down the list Enter a new value to over write a previous value and then press ENTER Remember that a minimum must be less than the corresponding maximum or the TI 92 will issue an error message Also remember to use the key not which is subtraction when you want to enter a negative value Figures 5 17 18 5 19 20 and 5 21 22 show different WINDOW screens and the corresponding viewing rectangle for each one TE TRE ae T m TT Figure 5 19 Square window Figure 5 20 Graph of y x 4x To initialize the viewing rectangle quickly to the standard viewing rectangle Figure 5 18 press F2 Zoom 6 ZoomStd To set the viewing rectangle quickly to a square Figure 5 19 press F2 Zoom 5 ZoomSqr More information about square windows is presented later in Section 5 2 4 Ce lzsonlrraceleraphitathorsule 2 A Figure 5 21 Custom window Figure 5 22 Graph of y x 4x am TPE
57. ss the symbol above the APPS key For now restore the TI 92 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 graph the table will not necessarily correspond to the graph unless you use TblSet to generate a new table based on the functions s being graphed as in Section 5 2 1 Runber 2 FE saps Bgoximare gt nieres co Tarr aE ea TaD ATER Figure 5 15 Split screen LEFT RIGHT Figure 5 16 MODE settings for Figure 5 15 Your graph window may look like the one in Figure 5 17 or it may be different Since the graph of y x 4x extends infinitely far left and right and also infinitely far up and down the TI 92 can display only a piece of the actual graph This displayed rectangular part is called a viewing rectangle You can easily change the viewing rectangle to enhance your investigation of a graph Figure 5 17 Graph of y The viewing rectangle in Figure 5 17 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 5 18 shows the WINDOW screen that corresponds to the viewing rectangle in Figure 5 17 This is the standard viewing rectangle for the TI 92 The variables xmin and xmax are the minimum and maximum x values of the viewing rectangle ymin and ymax are the minimum and maximum y values xscl and yscl
58. t to change You can also use F1 to go to page 1 or F2 to go to page 2 of the MODE menu To change a setting use to get to the setting that you want to change then press gt to see the options available Use A or V to highlight the setting that you want and press ENTER to select the setting To start with select the options shown in Figures 5 1 and 5 2 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 and approximate calculation mode Note that some of the lines on page 2 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 E Tar j FUNCTION a FULL nains HE TA fens PEAR 10 gt eo RADIANS ne f RORHAL gt RECTANGULAR gt Exact Hpprox Pretty Print ONS dahun 4 Enter SAVE ESC CANCEL gt H Enter SAVE gt Figure 5 1 MODE menu page 1 Figure 5 2 MODE menu page 2 Copyright Houghton Mifflin Company All rights reserved 2 Graphing Technology Guide TI 92 Technology Tip There are many different ways to get to the most commonly used screens on your T
59. t zoom factor is 4 in both direction It is not necessary for xFact and yFact to be equal Sometimes you may prefer to zoom in one direction only so the other factor should be set to 1 Press ESC to leave the ZOOM FACTORS menu and go back to the graph Pressing 2nd QUIT will take you back to the Home screen Technology Tip The TI 92 remembers the window it displayed before a zoom So if you should zoom in too much and lose the curve press F2 Zoom B Memory 1 ZoomPrev to go back to the window before If you want to execute a series of zooms but then return to a particular window press F2 Zoom B Memory 2 ZoomSto to store the current window s dimensions Later press F2 Zoom B Memory 3 ZoomRel to recall the stored window 5 2 7 Relative Minimums and Maximums Graph y x 4x once again in the standard viewing rectangle This function appears to have a relative minimum near x I and a relative maximum near x 1 You may zoom and trace to approximate these extreme values First trace along the curve near the local minimum Notice by how much the x values and y values change as you move from point to point Trace along the curve until the y coordinate is as small as you can get it so that you are as close as possible to the local minimum and zoom in press F2 Zoom 2 Zoomin ENTER or use a zoom box Now trace again along the curve and as you move from point to point see that the coordinates change by smaller amounts than before Keep
60. u where angle measure is selected Then press to display the options Use or Y to move from one option to the other Either press the number corresponding to the measure or when the measure is highlighted press ENTER to select it Then press ENTER to confirm your selection and leave the MODE menu It s a good idea to check the angle measure setting before executing a calculation that depends on a particular measure You may change a mode setting at any time and not interfere with pending calculations From the Home screen try the following keystrokes to see this in action Expression Keystrokes Display sin 45 MODE 7071067812 WV VENTER ENTER SIN 45 ENTER sin n SIN 2nd x 0548036651 ENTER sin x MODE o YYW MENTER ENTER SIN 2nd x ENTER sin 45 SIN 45 ENTER 8509035245 n SIN 2nd n 6 5 ma ENTER The first line of keystrokes sets the TI 92 in degree mode and calculates the sine of 45 degrees While the calculator is still in degree mode the second line of keystrokes calculates the sine of x degrees approximately 3 1415 The third line changes to radian mode just before calculating the sine of x radians The fourth line calculates the sine of 45 radians the calculator remains in radian mode The TI 92 makes it possible to mix degrees and radians in a calculation Execute these keystrokes to calculate tan 45 sin as shown in Figure 5 65 TAN 45 2nd MATH 2 Angle 1 SIN 2nd x 6 2nd MATH 2 Angle 2 ENTER Do
61. umber sub menu of the MATH menu Figure 5 8 From the Home screen calculate 6 78 6 by pressing 2nd MATH 6 floor 6 78 ENTER To graph y x go into the Y menu move beside y1 and press CLEAR 2nd MATH 6 ffoor X ENTER GRAPH Figure 5 26 shows this graph in a viewing rectangle from 5 to 5 in both directions The true graph of the greatest integer function is a step graph like the one in Figure 5 27 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 92 by going to the Y screen moving up until this function is selected highlighted and then pressing F6 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 fame ara Ti ae aE TRE Connected graph of y x Figure 5 27 Dot graph of y x Figure Technology Tip When graphing functions in the Dot style it improves the appearance of the graph to set xres to 1 Figure 5 27 was graphed with xres 1 Also the default graph style is Line so you have to set the style to Dot cach time you wish to graph a function in Dot mode Copyright Houghton Mifflin Company All rights reserved 12 Graphing Technology Guide TI 92 The TI 92 can graph piecewise define
62. ursor 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 92 retains as many entries as the current history area holds in a last entry storage area including entries that have scrolled off of 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 user s 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 the cursor pad 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 for the balance is P 1 where P principal r rate of interest expressed as a decimal n number of times interest is compounded each year and number of years In our example this b
63. ve Figure 5 40 Graph of y 25x Many different functions can be stored in the Y list and any combination of them may be graphed simultaneously You can make a function active or inactive for graphing by pressing F4 when the function is highlighted to add a check mark activate or remove the check mark deactivate Now go back to the Y screen and do what is needed in order to graph y1 but not y2 Now activate both functions so that both graphs are plotted Press F3 TRACE and the cursor appears first on the graph of y x 4x because it is higher up on the Y list You know that the cursor is on this function y1 because of the numeral 1 that is displayed in the upper right corner of the screen Press the up 4 or down Y direction to move the cursor vertically to the graph of y 25x Now the numeral 2 is shown in the upper right corner of the Copyright Houghton Mifflin Company All rights reserved 15 Graphing Technology Guide TI 92 screen Next press the left and right arrow keys to trace along the graph of y 25x When more than one function is plotted you can move the trace cursor vertically from one graph to another with the and Y directions Technology Tip By the way trace the graph of y 25x and press and hold either the or gt direction The cursor becomes larger and pulses as it moves along the graph Eventually you will reach the left or right edge of the window Keep pressing the direction and the
64. whose n th term is given by a enter values for the variables a d and r then evaluate the formula by pressing A R A N 1 ENTER To use the explicit formula in Seq MODE make u n 7 n 1 4 by pressing Y then using M to move up to the u1 n line and pressing CLEAR 7 N 1 x 4 ENTER 2nd QUIT Once more calculate w1 18 by pressing U 1 1 8 ENTER 5 7 3 Finding Sums of Sequences You can find the sum of a sequence by combining the features sum and 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 Jow to high For example suppose you want to find the sum Zao 3 Press 2nd MATH 3 LIST 6 sum 2nd MATH 3 LIST 1 seq 4 3 A K K 1 12 ENTER Figure 5 87 The seq Copyright Houghton Mifflin Company All rights reserved 32 Graphing Technology Guide TI 92 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 K could just have easily been an A or an N FeMlaigebralisiclotherProntolctear az t 3400 3 Now calculate the sum starting at n 0 by using gt and to edit the range You should obtain a sum of approximately 5 712848 5 8 Parametric and Polar Graphs 5 8 1 Graphing Par
65. will display the matrix T TE ig ia a fv ligebralCalclother Pranto clear a z 3 sa L 2a b 4 PRES amr ST TRE it Ear RET Figure 5 76 Matrix a Figure 5 77 Matrix b To add two matrices say a and b create b with the same dimensions as a and then press A B ENTER Again if you want to store the answer as a specific matrix say m then press STOP M Subtraction is performed in similar manner 2 0 3 Now create a matrix called c with dimensions of 2x3 and enter the matrix 5 as C For matrix multiplication of c by a press C x A ENTER If you tried to multiply a by c your TI 92 would notify you of an error because the dimensions of the two matrices do not permit multiplication in this way The transpose of a matrix is another matrix with the rows and columns interchanged The symbol for the transpose ofa is a To calculate a press A 2nd MATH 4 Matrix 1 ENTER Copyright Houghton Mifflin Company All rights reserved 28 Graphing Technology Guide TI 92 5 6 3 Row Operations Here are the keystrokes necessary to perform elementary row operations on a matrix Your textbook provides a more careful explanation of the elementary row operations and their uses eee largebralcaiclotner Pranto ciear a z E laigebralcaiclother Pramtolciear a z rouSwap a 2 3 2 0 4 6 ls mRowAidd 4 a 2 3 18 EEN o inRouhdd lt 4
66. y 1 3 gt x 4 The solution is the region which is below the graph of 1 2 and above x 4 First graph both equations Then from the graph screen press F5 Math C Shade The TI 92 will prompt for the function that you want to have the shading above Use or to move the cursor to the graph of x 4 then press ENTER The TI 92 will then prompt for the function that you want to have the shading below so use or W to move the cursor to the graph of 1 35 and press ENTER The TI 92 will then prompt for the lower bound then the upper bound which are the left and sik edges respectively of the extent of the shading If you do not enter a lower or upper bound the values of xmin and xmax will be used So in this case press ENTER twice to set the lower and upper bounds The shaded area extends left from x 2 hence the solution to Moana is the half line x lt 2 or 29 2 1 Copyright Houghton Mifflin Company All rights reserved 22 Graphing Technology Guide TI 92 Figure 5 64 Graph of 1 2x4 5 4 Trigonometry 5 4 1 Degrees and Radians The trigonometric functions can be applied to angles measured either in radians or degrees but you should take care that the TI 92 is configured for whichever measure you need Press MODE to see the current settings Press Y three times and move down to the fourth line of the first page of the mode men
67. y coordinate Figure 5 92 The rectangular coordinates are 3 0 flpigebralcaiclother Prantolciear a z ARAARA 7 elaigsbralcaiclotnerlpronto ciear a z lm RYPrC4 33 5 435011088 3 PRS m 3 PERSI o Figure 5 91 Rectangular to polar coordinates Figure 5 92 Polar to rectangular coordinates 5 8 3 Graphing Polar Equations The TI 92 graphs polar functions in the form r 8 In the Graph line of the MODE menu select POLAR for polar graphs You may now graph up to ninety nine polar functions at a time Be sure that the angle measure has been set to whichever you need RADIAN or DEGREE Here we will use radian measure For example to graph r 4 sin press Y for the polar graph editing screen Then enter the expression 4 sin 0 by pressing 4 SIN 0 ENTER The 8 key is on the lower right of the keyboard near the ENTER key Choose a good viewing rectangle and an appropriate interval and increment for In Figure 5 93 the viewing rectangle is roughly square and extends from 14 to 14 horizontally and from 6 to 6 vertically Refer back to the Technology Tip in Section 5 2 4 Copyright Houghton Mifflin Company All rights reserved 34 Graphing Technology Guide TI 92 Figure 5 93 shows rectangular coordinates of the cursor s location on the graph You may sometimes wish to trace along the curve and see polar coordinates of t
68. you get 1 5 whether your calculator is set either in degree mode or in radian mode The degree sign can also be entered by pressing 2nd D which saves keystrokes There is no corresponding key for the radian symbol Copyright Houghton Mifflin Company All rights reserved 23 Graphing Technology Guide TI 92 f airasia ren ances s sirf 2 ud tan 45 gt sin lt n 6 gt gt Han a APPAR INCI Figure 5 65 Angle measure Technology Tip The automatic left parenthesis that the TI 92 places after functions such as sine cosine and tangent as noted in Section 5 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 92 is in radian mode it calculates the tangent of 45 degree 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 92 is in radian mode it converts into radians rather than converting merely the 6 to radians Experiment with the placement of parentheses to see how they affect the result of the computation 5 4 2 Graphs of Trigonometric Functions When you graph a trigonometric function you need to pay careful sin 30x attention to the choice of graph window For example graph y in the standard viewing rectangle Trace along the curve to see where it is Zoom in to a better window or use th
69. zooming and tracing until you find the coordinates of the local minimum point as accurately as you need them approximately 1 15 3 08 Follow a similar procedure to find the local maximum Trace along the curve until the y coordinate is as great as you can get it so that you are as close as possible to the local maximum and zoom in The local maximum point on the graph of y x 4x is approximately 1 15 3 08 The TI 92 can automatically find the maximum and minimum points While viewing the graph press F5 Math to display the Math menu Figure 5 50 Choose 3 Minimum to calculate the minimum value of the function and 4 Maximum for the maximum You will be prompted to trace the cursor along the graph first to a point lefi of the minimum maximum press ENTER to set this ower bound Note the arrow near the top of the display marking the lower bound as in Figure 5 51 Puana Roey E eecn tracefkesrapnsin or sul Upper Boung x81 59663 yet 2 316316 RPE USE E ETE RD TEFCIECANCET a a AET Ta Figure 5 50 Math menu Figure 5 5 Finding a minimum Copyright Houghton Mifflin Company All rights reserved 18 Graphing Technology Guide TI 92 Now move to a point right of the minimum maximum and set a upper bound by pressing ENTER The coordinates of the relative minimum maximum point will be displayed see Figure 5 52 Good choices for the left bound and right boun

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