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TI-89/Voyage 200/TI-92ONLINE Graphing Calculator Manualfor Dwyer/Gruenwald’s

PRECALCULUSA CONTEMPORARY APPROACH

Dennis PenceWestern Michigan University

Brooks/ColeThomson Learning™

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Table of Contents

TI-89/Voyage 200/TI-92 Graphing Handhelds

Chapter 1 Foundations and Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Calculator Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Order of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Exponents and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Fractional Arithmetic and Exact Answers . . . . . . . . . . . . . . . 11Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Function Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Graphing a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Rational Functions and Vertical Asymptotes . . . . . . . . . . . . . 20

Chapter 2 Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Increasing and Decreasing, Turning Points . . . . . . . . . . . . . . 23Combinations and Composition of Functions . . . . . . . . . . . . 23Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Graphing a Family of Functions . . . . . . . . . . . . . . . . . . . . . . 24Piecewise-defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . 25Least-Squares Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 3 Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 28Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 4 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . 31Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Regressions Involving Exponentials and Logarithms . . . . . . 32

Chapter 5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Sine, Cosine, and Tangent Function Keys . . . . . . . . . . . . . . . 35Plotting the Sine, Cosine, and Tangent Functions . . . . . . . . . 36Families of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 37Cosecant, Secant, and Cotangent Functions . . . . . . . . . . . . . 37Plotting the Inverses of Sine, Cosine, and Tangent . . . . . . . . 38

Chapter 6 Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . . 38Graphical Check of Equations . . . . . . . . . . . . . . . . . . . . . . . . 38Conditional Trigonometric Equations . . . . . . . . . . . . . . . . . . 40

Chapter 7 Applications of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Complex Numbers Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 41Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Plotting Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 8 Relations and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Graphing Relations in Pieces . . . . . . . . . . . . . . . . . . . . . . . . . 46Implicit Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Plotting Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Plotting Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Plotting Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 9 Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . 49Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Identity Matrices, the Inverse of a Matrix, Determinants . . . 52Systems of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 10 Integer Functions and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 56Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Permutations, Combinations, Random Numbers . . . . . . . . . . 58

Note that in Acrobat Reader, each chapter and section in this table of contents is linkedto the appropriate location in the document. Click on an entry in this table of contentsto move to that place in the document. Similarly, chapter and section titles in thedocument are linked back to this table of contents. Web links are also active if yourcomputer has an internet connection.

TI-89/Voyage 200/TI-92, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc.

TI-89/Voyage 200/TI-92The TI-89 is a interesting choice for a graphing calculator to use while learning

from Precalculus. The older TI-92 and TI-92 Plus will do essentially the sameactivities presented here, but have a larger screen, full QWERTY keyboard, andgreater cost. The TI-92 has been replaced by the Voyage 200 with a smaller, lightercase, with more memory, with many “cost” applications included, and with theGraphLink cable included (at the same cost). All of the handhelds in this familyinclude a computer algebra system (CAS). (Texas Instruments is trying to differentiatethese machines from calculators since they can do so much more. Initially they tried tocall the Voyage 200 a personal learning tool (PLT). This phrase did not catch on, sothey now refer to them as handhelds.) Because these calculators can perform somealgebraic operations, they are sometimes banned in testing situations, so be sure tocheck to see if your instructor allows these. The Voyage 200 and TI-92 are notallowed in College Board testing (SAT and AP Calculus) because of the QWERTYkeyboard. The TI-89, Voyage 200, and TI-92 Plus have more memory and flashROM, enabling them to be electronically upgraded and to add further applications.You can look at the Texas Instruments graphing calculator web pages(http://education.ti.com) to find the latest information on free or commercial TI-89,Voyage 200, and TI-92 Plus applications that can be downloaded using a computerand the GraphLink cable. Also check for the newest operating system (OS) at thatweb site for the TI-83 Plus. A newer OS may fix problems and pave the way fornewer applications. The less expensive TI-89 is usually the choice in this familyunless you need to use the Geometry Application which works much better on thelarger screen and comes included in the other models. My preference is actually forthe Voyage 200, with the added cost easily justified by the GraphLink cable, increasedmemory, larger screen, and nicer keyboard. Unfortunately mass-market stores do nottend to carry the Voyage 200.

Chapter 1 - Foundations and Fundamentals

Calculator FundamentalsWhen you turn on a TI-89, Voyage 200, or TI-92, it usually comes up in the

HOME screen. If not (because the calculator did an “automatic shutoff” in anotherscreen), press " (for TI-89) or " (for others) to move to the Home screenwhere immediate computations are performed. [The newest OS allows these machinesto come on with a icon-based desktop. Select HOME for immediate computations.]

http://education.ti.com

Chapter 1 - Foundations and Fundamentals 6


Briefly, the labels at the top indicate menus or commands for the function keys (belowthe screen for TI-89/V-200 and to the left of the screen for TI-92). The middle area isthe “history” of recent commands and results. New things that you type will appear onthe command line near the bottom of the Home screen, and the small type at the verybottom indicates some mode settings and other information. While you are typing ona new command line (before Í), pressing ‘ will clear out everything in thecommand line to the right of the cursor. If there is nothing in the command line to theright of the cursor, pressing ‘ will clear out everything to the left. Pressing ƒand selecting 8:Clear Home will clear out old things in the history.

Icon Desktop (Voyage 200) Home Screen (TI-89)

Press z so that we can check (and explain) the various mode settings. Weshow the more extensive modes for a TI-89, Voyage 200, or TI-92 Plus (in a recentOS). Some of the items do not appear on the mode screen for the TI-92 (or older OS).

TI-89 Mode Screens: Page 1 Page 2 Page 3

Set your Mode Screen to agree with the figures here. (For a TI-92, ignore items whichdo not appear.) Under Exponential Format, Normal (the default) tries to show theentire number normally, but switches to scientific notation if a positive number is toolarge or too small. Sci always uses scientific notation, and Eng uses a specialscientific notation where exponents are a multiple of 3. Under Display Digits, FLOAT6 (the default) moves the decimal point or the scientific exponent to show 6 significant



digits (with zero suppression to the right). We can choose more or fewer significantdigits, or we can have a fixed number of decimal places displayed. Briefly, the otherlines on the first page specifies the graphing mode, the current folder, the angle mode,the complex number format, the vector format, and the fact that we want items in thehistory displayed in “pretty print” format. We will explore the other pages later, but itimportant to be in FULL rather than another style of Split Screen display for now. Youwill need to press ¸ once to confirm a new selection and again to save the newsettings in the Mode Screen. Under Exact/Approx Format, AUTO (the default) tries toshow the exact result, but switches to approximate floating-point notation if it cannot.

The keyboard layout for the smaller TI-89 is fairly simple. Pressing a key doeswhat is printed on the key. Pressing y (you do not need to hold it down) and thenanother key gives the operation printed above, left, and in the same color. Pressingƒ (you do not need to hold it down) and then another key gives the letter printedabove, right, and the same color (purple). Pressing the green diamond ¥ (you do notneed to hold it down) and then another key gives the operation printed in green abovethe keys. Some other keys have an unprinted “green diamond” command meaning,which you can see by pressing ¥^ The keyboard on a Voyage 200 or TI-92 is eveneasier because there is a separate QWERTY key area, eliminating the need for anALPHA shifting key. Note that the three different keys an a Voyage 200 or TI-92marked ¸ (and two different keys marked 2) have the same effect. Usewhichever one is most convenient. On both styles of keyboard you can get upper caseletters with the ¤ key. Since these calculators ignore case when dealing withcommands or variables, upper case letters are only nice for printed text of some kind.

While the catalog with all of the commands in alphabetical order is nice on any TIgraphing calculator, the catalog for this family has one additional feature. When youare looking at a command in the catalog and have it selected, then you can read a briefindication of the syntax for that command at the bottom of the screen. Select thecommand with ¸, and this syntax stays below the command line until you typesomething else.

TI-89 Catalog Voyage 200 Math Menu



The blue N key is used frequently to escape out of some dialog box or screenwithout making a selection. For example, you could escape out of the catalog withoutpasting a command on the command line using N. Similarly you could escape outof a menu without selection one of the options. The TI-89/92/200 family allows youto type commands character-by-character if you so choose. It is usually easier to getcommands from a menu or the catalog. Further submenus are indicated by an arrow.

Order of OperationCalculators generally follow the traditional algebraic order of operations. Note the

order of operation can be controlled with parentheses. This calculator allows impliedmultiplication (no multiplication symbol is needed between the two objects) in manysituations where there is no other interpretation. Just be careful with impliedmultiplication, because if there is any other interpretation possible, something else willhappen. Final parentheses cannot be omitted. The TI-89/92/200 family will report asyntax error if parentheses are not matched.

It is very important to recognize the difference between the subtraction key ¹

above the Í key and the negation key Ì one to the left in the bottom row of keys.In textbook notation we tend to use the same symbol for both, letting the contextdetermine the meaning. Notice on the screen that the negation is slightly higher andshorter. The subtraction operation takes two numbers as arguments, one before thekey is pressed and one after. The negation operation takes only one number as anargument coming after the key is pressed. If you start anew command line by pressing the subtraction key ¹, thecalculator assumes you wish to do a continuationcalculation. Thus it assumes that you want to subtractsomething (yet to be typed) from the previous answer.You can also get the previous answer anywhere withinthe command line with y [ANS] which is found abovethe negation key.



There are many situations where you want to execute essentially the samecommand repeatedly. There are some nice editing features that make this easy to do.The command y [ENTRY] found above the Í key causes the last command lineto be recalled so that you can edit it. Pressing y [ENTRY] several times allows youto go back to several previous command lines (limited bythe size of the history). When you edit a previouscommand line, you do not need to move the cursor pointto the end before pressing Í. If you want to executeexactly the same command line, you do not need to recallit. Just repeatedly press Í. In the screen shownhere, we have typed 11 Í and then pressed Ã 7

Í. As we repeated press Í, we add 7 to theprevious result.

There is also a simple way to store the result of a computation for later use. Thecommand is ¿ , and this command is represented on the screen as an arrow →.You follow this command by a variable name. Variable names must begin with aletter, ignore upper and lower case, and can contain at most eight characters. Thenwhen you need to use the result later, you simply type the variable name. You can andwill want to delete variables when you no longer need them to save memory. Thecommand DelVar in the † Other menu here in the Home screen followed by thevariable name will do this, or deleting can easily be donein the ° screen. It saves time if you store anintermediate computation rather than copying down anumber and retyping it later. Further, most people arelazy, and they copy down only a few of the decimalplaces. The “storing” operation saves the completenumber with all significant decimal places for later use.

Complex ArithmeticA TI-89/92/200 can handle complex arithmetic. Press 3 and select Complex

Format RECTANGULAR rather than REAL. The symbol for the imaginary ) is a secondfunction on the keyboard (above the ½ key on the TI-89 or above the letter “I”on the rest) . Do not try to use the regular letter I (either upper or lower case) on thekeyboard. Typing a number immediately before ) is one place where you can safelyassume implied multiplication. You can then add, subtract, multiply and dividecomplex numbers. The MATH menu, Complex submenu, has other commands forcomplex numbers.



The absolute value function in the MATH menu,Number submenu, has the traditional meaning for realnumbers. For a complex number, abs gives the modulus(or square root of the sum of the squares of the entries).In either case this result represents the “length” or “size”of a number and is always positive (unless the number iszero).

Scientific NotationEven in our Normal Exponential format mode, a number may be expressed in

scientific notation if it is too large. Calculators and computers have a short-hand forthis. Instead of printing out 5.7319 × 1025 which is difficult, they simply present5.731925. You should use the same short-hand when you want to enter a number inscientific notation (avoiding multiplication by a power of 10). Use the ^ keystrokewhere you want this symbol to be placed. Internally the calculator uses thisnotation, and 9.99999999999999 is the largest number it can handle. If acomputation results in a larger number, there will be a warning message and the

symbolic object ¸ will be used. 1⁻999 is the smallest positive number represented,and positive numbers smaller than that are rounded to zero.

Exponents and RadicalsThere are no special commands to square or cube a number. You must always use

the › key followed by the exponent you desire. This command also works for



negative and fractional exponents.

There is a special command for a square root (a 2 function on the keyboard abovethe multiplication key p). For other radicals, use a fractional exponent.

Fractional Arithmetic and Exact AnswersThe calculators in this family are capable of doing exact symbolic computations.

In the Exact/Approximate mode setting EXACT, they try to give only exact answers. Inthe setting APPROXIMATE, they always give answers witha decimal point and a limited number of significantplaces (often only an approximation to some exact resultsuch as1/7). The default setting AUTO tries to give exactresults if possible but switches automatically tonumerical approximations if necessary. If you want exactresults (such as for fractional arithmetic), make sure toenter numbers with no decimal points. If any number ina computation has a decimal point, then the calculator assumes it is an approximatevalue. It then gives the answer as an approximate value. Thus one way to force adecimal answer is to put in a decimal point. Another way to force an approximatedecimal answer is the press the green ¥ before the ¸ for a command line.

Scatter PlotsIt is possible to plot an individual point in the coordinate plane using the command

PtOn from the catalog. Issuing this command from the Graph screen, you get to selectthe point with the free-moving cursor (and Í). Issuing this command from theHome screen, you type the desired coordinates. Either way, the resulting point on theGraph screen is a drawn object that goes away if you resize the viewing window orregraph anything.

A more permanent way to plot several points is to use a statistical plot. Supposewe wish to plot the following data.



x 1.4 2.1 2.9 3.5 4.3

y 1.0 1.4 1.7 2.0 2.4

Press the blue O key and select 6:Data/MatrixEditor to move from the Home screen to the editorwhere we can enter this data. You get a choice ofworking with the current data (the last in the editor),opening another old data variable, or creating a new datavariable which is what we wish to do. Select the type asData, stay in the main folder, and type the variable nametest.

Type the x-data in column c1, pressing ¸ after each number. Then move overto column c2 up to the first row to enter the y-data. It is easy to move to any mistake,either typing over the wrong entry, inserting a new entry, or deleting an extra entry.The data is automatically “saved” as you type it. If you leave the data/matrix editor,this variable will contain the last things that you have typed. Also if you leave thiseditor, this data variable will become the current data set.

After the data has been correctly typed, press „ bring up the Plot Setup screen inthe Data/Matrix Editor. Press ƒ while Plot 1: is highlighted to define the statisticalplot details. We desire a Scatter style plot, using a Box as the mark, selecting the x-



data from column c1 and the y-data from column c2 as indicated.. Make sure to use¸ enough times to confirm entries in a given box as well as to save this plot setup.

Before we plot this, we need to set the viewing window, and we need to make surethat nothing else will appear in our graph. Press O and select 2:Y= Editor tomove to the function editor and make sure that no function formula is selected (byhaving a check mark to the left). Press † while highlighting the formula to toggle theselection check off if needed. Press „ Zoom to bring up some quick ways to resetthe window. For example, 9:ZoomData will always resize the window so that you cansee all of the data in a statistical plot. Here we have other reasons for preferring4:ZoomDec so that pixel coordinates come in even tenths. After getting the graph, wecheck to see what viewing window settings were fixed by pressing O and selecting3:Window Editor. Since these applications involving graphing are used so often, theycan also be obtained by pressing the green ¥ and the key with the green phrase aboveit.

Press O and select 4:Graph to move back to the graph. Pressing … Traceactivates some kind of tracing action in the plot. For a statistical plot, we can see thecoordinates of the points in the Scatter plot as we cursor right and left. Press †Regraph to have the plot redrawn (and to cancel the trace). In any graph, just movingthe cursor keys activates a free-moving cursor point in the plot. The coordinates ofthis free-moving cursor are displayed at the bottom of the screen. Now you see whywe like this “nice” viewing window.



Trace Estimate at x = 5.1

Suppose the reader is asked to estimate the y-value when the x-value is 5.1. Movethe free-moving cursor point to put in the approximate location we desire. It shouldalso be noted that statistical work (such as we have started above) is much easier usingthe optional (free) Statistical List Editor application. On most newer handhelds, thisapplication comes pre-loaded. For any machine with flash capabilities, thisapplication is available at the TI web site.

Function GraphingMake sure that the graphing mode is FUNCTION in order to graph a function of the

form y = some expression which is then typed in the Y= screen. PressO 4:Graph

ƒ 9:Format to get to the graphical format dialog box. Let’s make sure that all of ourfunction plots look the same by selecting the same formatting options here, matchingthe ones below. For example, let’s graph the function y = 3 x2

! 12 x + 14 in thestandard viewing window, as demonstrated in pages 54-55 of the text, by typing this inthe Y= screen. You get to this screen either by pressing O 2:Y= or by pressing¥ƒ. Clear out any other functions that may be stored there, and make sure that nostatistical plot is checked. The function key† in the Y= editor is used to turn a“check” on or off. Type the formula in slot y1 , press„ to zoom, and select6:ZoomStd as shown here.



Obviously this is not a particularly good choice for a viewing window for this functionas noted in the text. One can now set a viewing window to see this parabola in a littlemore detail. The zoom command A:ZoomFit resets ymin and ymax so that the graphjust fits with the screen for !10 # x # 10. Notice that we cannot see the x-axis anylonger because the setting for ymin is positive.

Select Zoom„, cursor down to B:MEMORY, and right to 1:ZoomPrev to get back to thegraph before the last zoom operation. Then try the first zoom command, 1:ZoomBox

to create a box around the parabola that nicely includes some of the axes for yetanother view.

There are many nice operations that can be performed while looking at a graph.The Trace… turns on a blinking pixel that can be moved right or left along the curve,showing the coordinates at the bottom of the screen. The x-coordinates are pixelcoordinates just as with the free-moving cursor, but the y-coordinates are actualfunction evaluations. Although we do not need it here, there are two nice ways to



change the viewing window while tracing. If you press Í while tracing, thewindow will shift so that the blinking pixel being traced moves to the center of theviewing window (called a Quick Zoom). If you trace all the way to the left or rightedge of the graph and then continue to try to go farther, the window will shift to letyou continue (called panning).

Pressing Math‡ and selecting 3:Minimum allows the estimation of the minimumof the function on a subinterval. You input a lower bound and a upper bound tospecify the subinterval (which can be done by just moving the cursor point slightly leftand right to where there is an apparent minimum).

Consider y = 0.018 x4! 0.45 x3 + 2.93 x2

! 1.5 x + 61.5 for 0 # x # 12.We get a plot as follows.

Tracing does not give integer x-values as we might want. As you are tracing, you canjust type any desired x-value or the Math command 1:Value allows us to evaluate thefunction exactly at a specific x-value such as an integer. Below we trace to anapparent maximum, use value to find the largest value at an integer, and use the Mathcommand 4:Maximum to explore this function.



By Trace with cursor By typing 6 while Tracing Maximum

Solving EquationsThere are several ways to solve equations when using this family of calculators.

We begin with the techniques available in the graphical screen. Consider the task ofsolving for the x-intercepts and y-intercepts for y = 1000 x3

! 15 x2 + 0.0002 fromExample 1.5.8 (page 73). We type the formula in the Y= screen and begin in thestandard viewing window with Zoom 6:ZoomStd as suggested in the text. Then weuse Zoom 1:ZoomBox several times to narrow in to a more appropriate viewingwindow as indicated below.

ZoomStd A more appropriate viewing window

Using the Trace is merely a crude way of approximating the x-intercepts. To get moreaccuracy, one needs to repeatedly zoom in. Instead, use the Math command 2:Zero tobegin a numerical routine to solve for the zero or root of this function. The routineasks the user to give a left bound and a right bound for the subinterval where youdesire to know the root. It is easy to give these bounds by moving the cursor point alittle to the left and right of the apparent zero on the graph and then pressing Í.



Repeat this procedure, giving different subintervals, to find the remaining roots.Finally the Math command 1:value followed by 0 displays the y-intercept.

Remember that this procedure will only locate an intercept contained within yourviewing window. The user might need to look at other larger viewing windows to beconfident that this function has no other intercepts outside the ones we haveconsidered. Panning and quick zooms might also help.

There is also a Math command 5:Intersect to numerically find an intersectionpoint for the graphs of two functions. Consider the two functions of Example 1.5.9(page 75), y = x3

! 7 x2 and y = 14 ! 17 x, in the given viewing window!2 # x # 8 and !60 # y # 30. This command prompts for the user to confirm whichtwo curves are desired and to specify a subinterval where the intersection is sought.



A second way to solve an equation is to use the application O 9:Numeric

Solver (which is not available on a TI-92). Simply type the equation in this interactivesolver and press ¸, give a guess for the variable to be solved (and a bound if youwish to restrict consideration to a subinterval), and press „ Solve. Note that theNumeric Solver leaves the computed value in the variable x (just in case we want touse it in the HOME screen). This will be a problem for us if we want to use x as analgebra variable. Thus we use the command NewProb to delete all single-lettervariables. In the HOME screen, the Algebra menu„, 8:nSolve command providesanother way to numerically solve equations (including on a TI-92).

The TI-89/92/200 family also has exact symbolic ways to solve equations.Remember to avoid using a decimal point as you enter number if you wish an exactanswer. The algebra menu in the HOME screen includes the command to solve anequation or to find the zeros of an expression. We can also use the factor command tofactor this cubic polynomial x3

! 7 x2! 14 + 17 x. .



Graphing a CircleWhen graphing a circle, it will look stretched or flattened unless the viewing

window is set so that a unit in the x-direction measures the same distance as a unit inthe y-direction. The Zoom command ZoomSqr will always change the viewingwindow to one with this equal scaling, adjusting either the pair {xmin, xmax} or{ymin, ymax} so that the new window includes everything shown previously.

Consider x2 + y2 = 64 , plotting the two functions andy x= −64 2

first in the standard window and then after Zoom ZoomSqr. Therey x= − −64 2

is also a [DRAW] 9:Circle( command to draw a circle, but drawn objects like thiscannot be traced.

ZoomStd Then ZoomSqr (and xres=1) Circle 0,0.8

Rational Function and Vertical AsymptotesThus far, we have been using the default Line style to get nice graphs of the

smooth functions considered. The calculator does this by plotting points (which arethe ones you see when you trace), and then by turning on other pixels in the plot tomake it look like those points are connected by short line segments. Most calculatorand computer plots work this way by default. For rational functions, this connectingof the dots leads to a deceptive picture. It is better to convert to the Dot graphing style

(or to at least look at both). Consider in the standard viewingyx x

=+

+−

−18

2

2

35

window. Notice that the near vertical lines at x = !2 and x = 3 appearing in the linestyle (where this function has vertical asymptotes) do not appear in the dot style.Notice that if we are only going to plot “dots”, the we will want xres = 1 to have moreof them. You do not notice the difference between xres = 1 or 2 in the line stylebecause the “dots” are connected with short line segments.

Chapter 2 - Functions and Their Graphs 21


Line Style (connected) Dot Style (and xres=1)

Chapter 2 - Functions and Their Graphs

Evaluating FunctionsAfter a function has been defined or a formula has been stored in the Y= editor,

there are several ways to calculate and display the value of the function. Whendefining functions in the HOME screen, we can use an “unassigned” variable as thefunction variable. For functions stored in the Y= editor, the variable must be x. Thesimplest way to evaluate a function is to calculate function values in the Home screen.

Consider from Example 2.1.14 (page 130).P v v v( ) .= +0 0178678 2.011683

First we define this in the HOME screen using the Define command from the Othermenu †. Notice function notation will take precedence over implied multiplication.

Notice that the solve operation gave a numerical result because the function formulahas floating-point numbers. We can also do function evaluation in the graphicalscreen.



We can also look at a table of values for a function stored in the Y= editor. In theTable Setup ¥&, we can choose between having the table entries automaticallygenerated using the tblStart and �tbl values or having the table entries determinedby asking the user.

We can even look at the graph and a table at the same time using a split screen mode.Only one part of the split screen is active at a time, and the keystroke 2a (abovethe O key) allows you to toggle back and forth between the two parts of the splitscreen. The graph and the table are not linked in any way, but they are based upon thesame evaluations and you can see both at the same time.

Now is a good time to mention the best way to choose a viewing window for a plot ofa new function. First put the formula for the function in the [Y=] editor. Then pressTable Setup [TBLSET] and set the tblStart and �tbl values so that we will get atable of function values where you think you want the interval [xmin, xmax]. Thethird step is to press [TABLE] to look at the function values. As you scroll through



these function evaluations, take note of how you will need to set [ymin, ymax] ifyou stay with the original idea about [xmin, xmax]. Often you will decide tochange even the x-interval as well after looking at a table of the function values. Thefourth step is to set [WINDOW] based upon what you have observed in the table. Finallypress [GRAPH] to see a plot that at least includes the pairs included in part of our table.

Increasing and Decreasing, Turning PointsWe can identify turning points and the subintervals in between where the function

is increasing or decreasing in a nice plot of the functions by using the graphical MATHcommands 3:Minimum and 4:Maximum while viewing the graph. Consider

from Example 2.2.8 (page 149). The graphs below are in thef x x x( ) = − +1

8

3 2 2

standard viewing window.

Combinations and Composition of FunctionsOnce we have typed several function formulas in the Y= editor, then we can work

with combinations and compositions without retyping, both in Home screen and in furtherfunction slots in the Y= editor. We can even use the symbolic capabilities of thiscalculator family to see formulas for the resulting combinations and compositions.Consider f(x) = 2 x2 + 4 x + 5 and g(x) = 2 x + 1, Example 2.3.4 (page 183).



Inverse FunctionsThe commands DrawFunc and DrawInv plot a non-interactive graph of a function and

the inverse of a function. Notice that this command for the inverse is really justinterchanging the x-coordinates and y-coordinates for plotting purposes. The functiondoes not need to be one-to-one and may not have a true functional inverse. Still the plotis correct when the function has an inverse.

Draw Commands Standard Window Square Window

Draw Commands Standard Window Square Window

Graphing a Family of FunctionsA quick way to plot several functions in a family is to use a list of numbers in place

of a single number as a parameter in the formula for the family. For example, we can seethe functions in the family f(x) = a x2 which are plotted in Figure 2.71 (page 183) byusing the list {-2, 0.5, 1, 4} in place of the parameter a.



Piecewise-defined FunctionsPiecewise-defined functions cannot be handled on a TI-89/92/200 using logical tests

at a multiplicative factor. A logical test on a TI-89/92/200 evaluates to “true” or “false”,not a numerical value. Instead, there is the special command when( to define a functionwith two pieces. Consider

3 26 9 4, 3( )

3, 3

x x x xf x

x x

− + − + <=

− ≥

from Example 2.5.5 (page 187). The required command line isDefine f(x)=when(x<3,-x^3+6x^2-9x+4,x-3).

Notice that in the line style, the nearly vertical line between dots connects the two pieceswhere it is not appropriate. The dot style does not do this (although it also leaves dotswithin pieces unconnected as well). To improve both plots, we have changed the windowvariable xres to 1.

Line Style Dot Style



It is possible to nest the when commands to get more than two pieces, but this quicklygets very confusing. Far better is to program a piecewise-defined function with severalpieces as a function type program. Consider

g x

xx

x x

xx

( )

, ,

, ,

,

=+

< −

− − ≤ <

−

+≤

R

S

||

T

||

3

11

2 1 1

3

11

2

2

plotted in the Zoom ZoomDec viewing window with the Dot Style.

y1(x)=g(x), Dot Style

Least-Squares Best FitThe TI-89/92/200 provides several different regression fits for numerical data,

including using linear, quadratic, cubic, and quartic polynomials. We demonstrate hereonly a linear fit. Consider Table 2.10 (page 208) giving U.S. health-care expenditures (inbillions of dollars) for a range of years.

Year 1985 1990 1995 2000

U.S. health care expenditures 422.6 666.2 991.4 1,299.5



The textbook suggests that you might want to convert 1985 to t = 0, 1990 to t = 5, etc.The purpose of this is to make the numbers smaller (which is usually nicer for handcomputations). Here we show that there is no need on the calculator to do this. Thus ourregression function will be different (having a different definition of the variables). Ourgraph will have the actual years as the first coordinate, and to evaluate the regressionfunction for 2003, we will simply need to enter in the variable 2003 (not t = 18) Enter theyears and the expenditures in data set in the Data/Matrix Editor. As we did before inChapter 1, turn on a statistical scatter plot of this data and use Zoom ZoomData to size theviewing window in an appropriate manner for this data.

Data/Matrix Editor, New F2 Plot Setup, F1 for Plot 1

Get back to this data set in the Data/Matrix Editor (now selecting Current) and press Calc‡ and select calculation type 5:LinReg, x-list c1, y-list c2, and have the RegEQ storedin y1(x). Below we have plotted the line on the same graph as the scatter plot.

Data/Matrix Editor, Current LinReg Results Scatter Plot with y1(x)F5 Calc, Store RegEQ in y1(x)

In a statistics course you will learn to interpret the significance of the correlation and r2.We will simply note that when the correlation is nearly 1, the linear regression line is arelatively good fit to the data. Notice that our result is

y1(x) = 59.118 x + -116947.69which does not agree with the E(t) = 59.118t + 401.54 given in the text. When we use theformula for a prediction for the year 2003, we do get the same result.

Chapter 3 - Polynomial and Rational Functions 28


y1(2003) = 59.118 (2003) !116947.69 = E(18) = 59.118 (18) + 401.54 = 1465.664

Again we note that most people find that they like to do statistics better using the freeflash application called the Statistical List Editor on the TI-89/92 Plus/200 family. Thiswill come loaded on a Voyage 200 (and on most new TI-89's) and is available from theTI web site. The process uses lists instead of the data structure above.

Chapter 3 - Polynomial and Rational Functions

Polynomial FunctionsA graphing calculator is very nice for investigating polynomials of degree three or

higher. We use the same techniques for setting viewing windows, finding zeros, andfinding turning points as for other functions. The added feature regarding work withpolynomials is that we have a few theorems to help us know when we have found enoughzeros or turning points. Further the TI-89/92 family of calculators has a few symbolicoperations for polynomials and rational functions.

Here is one trick for making the evaluation of a high degree polynomial more accurateand the graphing of it more rapid. Algebraically we canrewrite a polynomial in several equivalent ways. Forexample,p(x) = 3 x5

! 2 x4 + 7 x3! x2 + 4 x + 6

= ((((3 x ! 2) x + 7) x ! 1) x + 4) x + 6.If we key the second way into a calculator rather than thefirst, we can avoid using the “power key” ›, which is quiteslow for repeated computations, and we reduce the totalnumber of arithmetic operations required in evaluation. On virtually all graphingcalculators, entering a fifth degree polynomial in the second way will cause it to plot inabout half the time as entering it the first way.



Polynomials can get very big, making the Zoom A:ZoomFit seem an attractive optionafter you have set the x-interval for the desired window. We use this to plot the abovefifth degree polynomial (with an initial x-interval the standard [!10, 10]).

While this shows the end behavior fairly well, it often leads to some ridiculously large y-values. Let’s try the more reasonable process of looking at a table of values first beforesetting the first viewing window.

Considering evaluations for x between !10 and 10 by scrolling down the table (startingwith tblStart = !10 and )tbl = 1), we come to the following more reasonable window.We would then probably want to use a ZoomBox to investigate the zero a little morecarefully.

ZoomBox around zero

A TI-89/92/200 graphing calculator has special features for symbolic operations withpolynomial multiplication, division, or complex roots. In the HOME screen, look in the



Algebra menu for commands to solve equations, factor polynomials, find zeros offunctions, and (in the Complex submenu) commands to do these operations usingcomplex numbers

(cSolve, cFactor, cZeros).Consider the polynomials from Examples 3.3.1 (page 245) and 3.3.6 (page 251).

f(x) = 5 x4 + 8 x3! 29 x2

! 20 x + 12g(x) = x5

! 2 x4! x3 + 4 x2

! 2 x ! 4

We get exact factoring of f(x) = (x !2)(x + 1)(x + 3)(5x !2) and exact zeros {!3, !1,2/5, 2}. We get the (real) factoring of g(x) = (x ! 2)(x + 1)2(x2

! 2x + 2), the complexfactoring g(x) = (x ! 2)(x + 1)2(x + (-1) + i)(x !(1 + i), and the complex zeros {!1, 2,

, }.1 i− 1 i+For larger degree polynomials, the free flash application

called the Polynomial Root Finder will nicely findnumerical approximations for real and complex roots. Wedemonstrate here with g(x). Note that any numerical rootfinding algorithm will have trouble with a double root.Here the double root !1 is approximated by two complexroots with very tiny imaginary parts. This is not a mistake,but simply the result of the fact that when we round in the numerical computations, weeffectively get the roots of a slightly different polynomial. Thus for multiple roots, wewill prefer symbolic attempts to find the roots.

Chapter 4 - Exponential and Logarithmic Functions 31


Rational FunctionsFor a detailed look at vertical and horizontal asymptotes for rational functions, it is

convenient to zoom in and out in one direction at a time. Also don’t forget that the dot

style generally is best for this family of functions. Consider fromf xx x

x x( ) =

− +

−

2

2

2 2

2 4Example 3.5.4 (page 274) on various windows .

!3 # x #5, !5 # y #5 !25 # x #25, 0.4 # y #0.6 1.9 # x #2.1, !50 # y #55Overall view Highlighting end behavior Vertical view near x = 2

Chapter 4 - Exponential and Logarithmic Functions

Exponential FunctionsWe can nicely plot the family of exponential functions of the form f(x) = ax using

a list for a to reproduce Figure 4.4 (page 296). Try the trace on this plot.

Only the natural exponential function ex is provided on the keyboard, and you shoulduse this rather than the power key for more accuracy. You can use the natural exponentialkeystroke to get the value of this number e with e^(1) 2.71828182846. Using thesemethods will be better than typing these digits because even the guard digits (90) youcannot see will be correct.



Logarithmic FunctionsOnly the natural logarithmic function, x on TI-89 or µ on TI-92/200, is provided

on the keyboard. A common logarithm command log can be found in the catalog. Usethe Change-of Base Formula (page 326) to work with logarithms in another base in termsof one of these special ones.

loglog

log

ln

ln, ,

au

u

a

u

aa u= = ≠ >1 0

Regressions Involving Exponentials and LogarithmsThe Data/Matrix Editor statistical Calc menu offers a number of regression options

that involve families of exponential and logarithmic functions. The preliminary steps forthese regressions are the same as for linear regression above in Chapter 3. You simplyselect and plot a different regression fit. You can even plot several on the same screenand decide visually which seems to be the best fit.

LnReg a + b ln(x)ExpReg a bx

PowerReg a xb

Logistic (only on TI-89/92 Plus/200)a

b ed

c x1++

Logist83 (in Statistics/List Editor app)1 b x

c

a e−+

For example, consider the data from Table 4.7 (page 344) describing a state deerpopulation (in thousands) since 1994 (t = 0). We show how to obtain an exponential fitfor the data.



Year (since 1994) 0 1 2 3 4 5

Population (in thousands) 10,000 11,500 13,200 15,100 17,400 20,100

Type the data into a data set and obtain a scatter plot. Thenperform the exponential regression, and save the regressionequation in a function slot. Finally, compare the scatter plotto the graph of the regression equation. All of this can bedone in the Data/Matrix Editor for a data structure as wehave done before for linear regression. Here we move tothe free flash application Statistics List Editor to do theexponential regression instead. I simply find the erroranalysis to be easier in that editor.

First I highlight the name of an old list (that I no longer want), and then I press ¸ M ¸ to empty out the list for use in this work. (Or I create a new list namefor this work.) Then I proceed as indicated below.

In the above screens, we have stored the RegEqn in y1(x) = 9992.4074 (1.1495069)x. Inthe graph, we have evaluated y1(6) = 23017.567 to get the prediction from the model. We

can also rewrite the model function in the form .( ) btP t a e=

Chapter 5 - Trigonometric Functions 34


( ) ( )( )( ) ( )

ln 1.1482069

ln 1.1482069 0.13907203

( ) 9992.4074 1.1482069 9992.4074

9992.4074 9992.4074

tt

t t

P t e

e e

= =

= =The logistic regression is a very difficult computation. The routine in the TI-89/92

Plus/200 may fail to converge. It may have problems with large data, but in the mostrecent version it was successful for this data.

( )0.6330975

15459.8442( ) 8645.7941

1 9.0160715 xy x

e−= +

+For the TI-83 style logistic model (which better matches the text), the regression failedto be successful, even when we divided list2 by 1,000.

Chapter 5 - Trigonometric Functions

Angle MeasurementThe TI-89/92/200 has an angle mode setting of either RADIAN or DEGREE in the mode

screen. Note that the angle mode settings normally appears in the bottom line in smallprint of the HOME screen. We will experiment here with both settings. Pressing I2:Angle brings up a menu with further angle commands. The first 1:° causes thenumber before this symbol to be interpreted as degrees, regardless of the angle mode. Thesecond 2: gives radians, again regardless of the angle mode. The single quote anddouble quote symbols are 2 keystrokes found above Á and ¨ on the TI-89 or abovethe “B” and “L” on the TI-92 Plus or Voyage 200. Note that there is also a 2 keystrokefor Ä above the › key.



Assuming degree mode setting, expressions given in degrees-minutes-seconds (DMSnotation) will be converted to decimal degrees. The command I 2:Angle 8:DMS

converts something in decimal degrees into DMS. Expressions designated in radians withwill be converted to degrees. In degree mode, the degree symbol ° alone does nothing.

Assuming radian mode setting, expressions given in degress or degrees-minutes-seconds (DMS notation) will be converted to radians. The command y [ANGLE]

4:DMS still converts something in decimal into DMS, interpreting the decimal as decimalradians. Expressions designated with only the degree symbol ° will be converted intoradians. In radian mode, the radian symbol does nothing.

Sine, Cosine, and Tangent Function KeysThe keystrokes WXY or ˜ ™ š interpret their argument based upon

the angle mode unless a degree or radian symbol is present to override the angle mode.Being able to override the angle mode should mean that you do not need to change amode setting to switch back and forth between degrees and radians for simpletrigonometric computations. The most common error made when working with thesefunctions is to be in the wrong angle mode. A goal should be to know enough abouttrigonometric functions so that you can immediately recognize when you start to getanswers appropriate for the wrong angle mode. Note that the trigonometric keystrokescome with a left parenthesis (and expect you to type the right parenthesis). The calculatorwill try to give exact results unless you press ¥¸ or use a decimal point to force anapproximate answer.



Degree Mode Radian Mode

Degree Mode Radian Mode

The inverse trigonometric functions [SIN-1] [COS-1] [TAN-1] also depend upon theangle mode, not for the argument but for the output. There is no way to override this.Thus if you desire to interpret the answers from these inverse trigonometric functions indegrees, you must be in degree angle mode.

Plotting the Sine, Cosine, and Tangent FunctionsSince graphing calculators are used to plot trigonometric functions so often, a special

viewing window is provided that is frequently appropriate for these functions. Thecommand Zoom 7:ZoomTrig resets the viewing window to

Degree Mode !592.5 # x # 592.5, xscl = 90, !4 # y # 4, yscl = 1.

Radian Mode− ≤ ≤ =RST − ≤ ≤ =

10 3410758181 10 3410758181 2

4 4 1

. . , ,

, .

x Xscl

y Yscl

π

The unusual endpoints for the x-interval give nice fractions of 90° or radians as pixelπcoordinates for tracing. Below are examples in radian angle mode.



Sine Cosine, Thick Style Cosine, Thick StyleTangent, Line Style Tangent, Dot Style

Families of Trigonometric FunctionsWe can plot several functions in a family again by using a list for one of the

parameters. First we create a list par ={0.5, 1, 2, 4}. Then we store 1 in variables a, b,and c. One at a time, we replace a letter by the list to see the effect on the graph of

f(x) = a sin(b x + c).

y1(x) = par *sin(b*x + c) y1(x) = a sin(par*x + c) y1(x) = a *sin(b*x + par)

Cosecant, Secant, and Cotangent FunctionsThere is no keystroke for the remaining trigonometric functions on the TI-89/92/200.

You can get these knowing the fundamental identities for how csc x, sec x, and cot x arerelated to sin x, cos x, and tan x (namely that they are reciprocals).

For csc x type sin(x)^-1 or 1/sin(x).

For sec x type cos(x)^-1 or 1/cos(x).

For cot x type tan(x)^-1 or 1/tan(x).

However if you have OS 2.08 or higher (sorry but no upgrades for the old TI-92) datingto March 2003 or later, you have the other trigonometric functions and their inversesincluded in the catalog.

Chapter 6 - Trigonometric Identities and Equations 38


y1(x) = sec(x), Dot Style; y2(x) = sec-1(x), Square Style; ZoomTrig

Plotting the Inverses of Sine, Cosine, and TangentAgain, the definition of these functions and what you get when you plot them depend

upon the angle mode setting. Assume here radian angle mode. The command Zoom

7:ZoomTrig still gives a reasonable viewing window, although we may only be using asmall part of it. Notice if you try to trace to an x-value where the function is not defined,you lose the blinking pixel and no y-value appears.

y1(x) = sin-1(x); y2(x) = cos-1(x); y3(x) = tan-1(x); ZoomTrig

Chapter 6 - Trigonometric Identities and Equations

Graphical Check of EquationsWhen first presented with a trigonometric equation, a graph is one tool that we can

use to investigate whether the equation is an identity, a conditional equation, or acontradiction. Generally we graph the two sides of the equation separately and look forintersections. When you trace, use the up and down cursor keys to toggle between the twodifferent sides. It is even more convincing for identities to look at a side-by-side table ofevaluations of the two sides.



Example 6.1.1 (page 462)

2 sin x = 2 - 2 cos x

Example 6.1.2 (page 463)

(sin x + cos x)2 = 1 + sin 2x

Example 6.1.3 (page 463) 2 - sin x = cos x



Conditional Trigonometric EquationsWe have a variety of tools to use for solving conditional equations. We demonstrate

these on the equation cos 2x = 2 cos x from Example 6.4.9 (page 501). If we have firstplotted both sides, then we can compute intersections of the two separate curves in thegraph. Just make sure that your subinterval is very close to the intersection you want.

y1 = cos 2x, y2 = 2 cos x y3 = 2 cos2 x - 2 cos x - 1

If we rewrite the equation so that one side is zero, we can seek a zero on the graph of thefunction represented by the non-zero side instead as in y3 above. Finally, if we manage

to reduce the problem to something such as , then we can usecos (1 3) 2x = −[COS

-1] and our knowledge about the reference angles to solve for x in the interval

[0. 2B).

We also have some symbolic operations to use including a trigonometric expand andcollect, texpand and tcollect. Notice that the exact solution for a trigonometricequation will generally indicate the multiple solutions with the symbol @n{wholenumber} to stand for an arbitrary integer.

Chapter 7 - Applications of Trigonometry 41


, exactly or1 13 1 3 12 sin or 2 sin

2 2 2 2x n x n

π ππ π− − − −= − − = + +

, numerically.2 1.94553 or 2 1.94553x n x nπ π= + = −

Chapter 7 - Applications of Trigonometry

Complex Numbers RevisitedThe rectangular form for representing a complex number is a + b i, and there is a

complex format mode setting RECTANGULAR to enable this on a TI-89/92/200. You canalso find the symbol i as a y keystroke (above ½ for TI-89 and above the letterI for the rest). The trigonometric form for representing a complex number ( from page538) is r (cos 2 + i sin 2). This is available too in a slightly different notation calledthe POLAR form r ei2 . The variables r and 2 have exactly the same meaning in thetrigonometric and polar forms. In fact, the definition of a complex exponential e"+ i $ =e" (cos $ + i sin $) quickly reduces to ei 2 = cos 2 + i sin 2 . (See Exercise 7.4.51 onpage 555 for more detail about what is called Euler’s formula.)

Note that the angle mode (radians or degrees) affects how the angle2will be given in thepolar form.



Degree Angle Mode Radian Angle Mode

Note that you can type in complex numbers in any form. Often the resulting complexnumber is too long to see all of it on the screen at once. At any time, just press the upcursor to move to the history part of the HOME screen, and scroll right and left to see theresult. The modulus is obtained by the command abs.

The square root command and the power command (toget other nth roots) give principal roots (not all nth roots).For example, the fifth roots of 3 e1.2 i can be found from theprincipal fifth root x = 1.24573094 e0.24 i = r ei 2 given bythe calculator by repeatedly adding 2B /5 to the argument2.Thus we get the collection of fifth roots to be

.re re re re rei i i i iθ θ π θ π θ π θ π

, , , ,+ + + +2

54

56

58

5c h c h c h c h{ }Note that the calculator program in Exercise 7.4.50 (page 554) will not run as writtenthere. (The old-fashioned command IS>(K,N) indicating to “Increase K by 1 but Skipthe next command if K > N” has not been left on the newest handhelds.) However thereare more modern commands for looping. Type in the following Program-type Programto do the same thing, where we input the number of sides as a parameter in the programcommand.



Polar CoordinatesOn a TI-89/92/200, conversions between rectangular coordinates and polar

coordinates are implemented as commands in the MATH menu, Angle submenu. Thecalculator has chosen to convert the rectangular (0, 0) to the polar (0; 0). It gets a uniquepolar representation for rectangular coordinates other than the origin by selecting r > 0,0 # 2 < 2 B.

In the Graph formats screen, you will find the first formatting option is to selectrectangular graphing coordinates RECT or polar graphing coordinates POLAR. This formatoption will determine the coordinates that appear at the bottom of the graphical screen inall graphing modes.

Plotting Polar EquationsOn the 3 screen, move from function graphing to polar graphing by selecting

POLAR. On the % ƒ Tools 9:Format screen, select polar graphing coordinatesPOLAR. Then press # to see the polar equation editing screen. Type in the formulas for

r = 2 cos 2 and r = 1 + 2 sin 2from Example 7.5.7 (page 561). The graphing variable is now2 so it can be obtained bypressing ¥Ï above the key Z (or just the Ï key on the rest).



In addition to setting the x-range and y-range on the Window Editor, you now must alsoset values for the polar graphing variable 2. A good initial choice is to try an interval of[0, 2B] for 2, although this may not always be best. Here we choose Zoom 4:ZoomDecto get a decimal, equally scaled viewing window and also to get [0, 2B] for2, with2step= B/24 so that we hit favorite multiples of B as we trace.

The ‡ Math menu no longer contains an “intersect” command, and there is a goodreason for this. An apparent point of intersection of two polar equations can occurbecause of one representation of that point in one equation and a different representationof that same point in the other equation. For example, the two polar equations plottedabove appear to have three points of intersection. By tracing to find the approximatepolar coordinates giving the point on each equation and by turning on rectangulargraphing coordinates as well, we can roughly compute the following table describingthese three points and how they solve each equation.

(x, y) r = 2 cos 2 r = 1 + 2 sin 2(1.6, 0.8) (1.8, 0.4) or (-1.8, 3.5) (1.8, 0.4)(0.3, 0.7) (0.8, 1.2) or (-0.8, 4.3) (-0.8, 4.3)(0, 0) (0, 1.57) or (0, 4.7) (0, 3.7) or (0, 5.8)

We can get more accuracy on a TI-89/92 Plus with the interactive Numeric Solver, usingthis initial graphical work for starting guesses and for setting the equations to be solved.



VectorsThere is no special vector data type, but there are many special vector operations

which can be found deep in the MATH menu, I 4:Matrix L:Vector ops.Included in this Vector ops sub-submenu are the commands to convert betweenrectangular and polar coordinates (in vector form). To use these operations we need toenter vectors as a 1×2 matrix, or a 2×1 matrix. You can use the Data/Matrix Editor, oryou can type vectors in the HOME screen between square brackets separating theelements of the vector by commas. Vector addition, vector subtraction, and themultiplication of a vector by a scalar can be computed. Included in the mathematicalfunctions and operations for vectors are commands for finding a unit vector in the samedirection as a given vector and for finding the norm of a vector.

It is possible to use the drawing command for a line segment to get a rough sketch ofthe magnitude of a vector and to picture the idea of one vector added to the end ofanother. The command is Line and it expects as argument the coordinates of the startingpoint and ending point. Unfortunately there is no simple way to put an arrow at the end

Chapter 8 - Relations and Conic Sections 46


of any of the line segments to indicate direction. Here we draw line segments to represent

P(-1, -2), Q(-3, 1), and with the initial point of each vector at the origin. Then wePQ

add another line segment for putting the initial point at the terminal point for P(-1,PQ

-2) using the command Line ⁻1,⁻2,⁻3,1. Our viewing window is from ZoomDec andwe have RECT selected for formatting graphing coordinates so that the free-moving cursorcan help us locate endpoints.

ZoomDec Window

Chapter 8 - Relations and Conic Sections

Graphing Relations in PiecesThe TI-89/92/200 handhelds can graph a general relation with two variables, but it

is not simple to do. We begin using the easier technique where we solve the equation fory (possibly with more than one solution or piece). Looking at Example 8.1.9 (page 600),

we solve Just to highlight some potential4 9 36 36 4 92 2 2x y y x+ = = ± −as b g / .

difficulties, we select the window with ZoomTrig and plot both the upper (+) and lower(-) parts of the ellipse as separate functions (in function graph mode, RECT graphingcoordinates).

ZoomTrig Window



Notice that the upper and lower parts of the ellipse do not quite meet. Each of thesefunction formulas is defined only for !3 # x # 3 . As we move right with a trace point,we find the largest x-pixel coordinate to plot is x = 2.87979.The next pixel to the right has coordinate x = 3.01069, andin this column of pixels there is no plot. We do not landexactly on x = 3 as a pixel coordinate, where both y1 and y2would evaluate to zero. Using ZoomDec on the right heredoes give such a pixel coordinate at all of the integers aswell as other decimal values.

Implicit PlottingWe briefly look at how a special format style of 3D

graphing can be used to plot more general relations.Usually in 3D graphing mode, we plot function of the formz = f(x, y), giving a viewing box a # x # b, c # y # d, zmin# z # zmax. The simplest 3D style is a WIRE FRAME, withxgrid and ygrid determining the fineness of the mesh forthe wire frame. We show the following wire frame plot tomake this a little more clear. Consider z = (x2 + y3

! 10)/50 in the standard Zoom

ZoomStd box !10 # x # 10, !10 # y # 10, !10 # z # 10.

ZoomStd Box

The 3D graphing format style IMPLICIT PLOT shows the xy-plane where the surfacez = f(x, y) intersects that plane (i.e. z = 0). Thus we see a plot of the relation

x2 + y3! 10 = 0.

Be prepared to wait (you see the percentage completed as you wait) because these implicitplots take a long time. Making xgrid and ygrid larger will improve the plot quality (buttake longer to finish).



!10 # x # 10, !10 # y # 10Plotting Parabolas

A parabola that opens upward or downward is easily plotted as a single functionbecause we can solve uniquely for y in the equation. For a parabola that opens right orleft instead, we can either plot two separate functions (where we can trace on each piece)or we can switch the variables x and y and use the DrawInv command. From Example

8.2.7 (page 619) consider The plots below are in a standardx y+ = − −( )1 21

2

2.

viewing window.

Tracing Function Free-moving Cursor Near Drawn Object

Plotting HyperbolasIn all cases, a hyperbola will need to be plotted as two pieces in function graphing

mode. When the transverse axis is horizontal, we will face the problem of the two piecespossibly not meeting because of pixel coordinates not exactly hitting the vertices.Consider the hyperbola described in Example 8.4.3 (page 646).

Chapter 9 - Systems of Equations and Inequalities 49


Plotting Parametric EquationsA TI-89/92/200 handheld can nicely plot parametric equations. We demonstrate this

using x = 3 cos t ! 2, y = 5 sin t + 1 from Example 8.6.5 (page 671). In parametricgraphing mode, the letter t gives the graphing variable t. Below, we started first with thestandard viewing window ZoomStd, and then did a ZoomSqr to get equally scaled axes.

Notice that when we trace, we can see the value of the parameter t as well as the x-and y-coordinates of the point highlighted. Pressing theright cursor key increases the value of the parameter t(which will not necessarily cause the point to move right).While you are tracing, you can also type a desired t-value.The window settings have changed for parametric equationsas well. We set the t-interval for the parameter as well asthe bounds for the axes. The setting tstep determines theplotted points (which can then be traced). In the connectedgraphing mode, small line segments are drawn between the plotted (traceable) points. Iftstep is too large, these line segments may not be small, and our plot may be rather crude.If tstep is too small, it will take a long time to plot the parametric equations.

Chapter 9 - Systems of Equations and Inequalities

MatricesWhile you can enter very small matrices in the HOME screen, it is more convenient

to use the Data/Matrix editor. After you select this application from the O menu, youchoose New, select type Matrix, select a folder (usually main), give a variable name,specify the number of rows and columns of the matrix. Below we create three matricesand show how to do simple matrix arithmetic. In general you will find matrix operationsin the MATH menu, Matrix submenu.



Error Message Highlight Result ¸ to Command Line

An error message will appear if you try to add, subtract, or multiply matrices which do nothave the correct dimensions. The command to augment allows you to create a “wider”matrix by combining two matrices with the same number of rows. In particular, thiscommand can be used to form the augmented matrix using the coefficient matrix and theright-hand side of the equation. The square brackets can be used in the home screen tocreate small matrices.



Gaussian EliminationAll of the elementary row operations are provided. MATH menu,4:Matrix submenu,

J:Row ops sub-submenu. Generally when you execute one of the elementary rowoperations, you will want to store the result in some matrix slot. If you want, you canstore successive results in the same matrix, overwriting the previous information as wedo below. Or you can store results in a new matrix name.

If we follow a matrix name by the row and column in parentheses, we can isolate anindividual entry in the matrix. There is also a command to get the dimension of a matrix(with the result being a list containing the two dimension numbers). Random matricescan be generated by specifying the size, and they have single digit integer entries.

You can also have the calculator do the complete Gaussian elimination process on amatrix. The command is ref( to convert to a row-echelon form equivalent to the startingmatrix. Gauss-Jordan elimination is done by the command rref( to convert to the uniquereduced row-echelon form equivalent to the starting matrix. If the result is too large to



view at once, scroll right or left to see all of it before beginning the next command line.

The free flash application Simultaneous Eqn Solver canalso solve a system of linear equations. This solver caneven handle non-square systems (with many solutions).Normally you will just type the coefficients within thesolver, but you can also get the coefficients and the “right-hand-side” vector from a matrix.

Identity Matrices, the Inverse of a Matrix, DeterminantsYou can quickly get an identity matrix (with ones on the diagonal and zeros

elsewhere) with the command MATH Matrix 6:identity( by simply giving the sizedesired for this new square matrix. For a square matrix which has an inverse, the key—gives the inverse.



We have three ways (other than the flash application Simultaneous Eqn Solver) tosolve a system of linear equations such as

2 3 6

4 3

11

2

x y z

x y z

x y z

+ + =

− + = −

+ + =

involved in Figure 9.16 (pages 749). One way is to form the augmented matrix [A|B] andapply rref to it. The second way is to find the inverse A-1 of the coefficient matrix A andmultiply it times the right-hand side B. We can also find the determinant, say of matrixA. The third way is the command simult( found in the MATH menu, Matrix submenu.If you have the

Systems of InequalitiesConsider Example 9.7.4 (page 772) which asks for a graph of this system of

inequalities.x y

x y

+ ≥

− + ≤

2 2

3 4 12

Enter each inequality as a function equality solved for y, and select the style (shade aboveor shade below) to match Figure 9.30 (page 772) using the window !9 # x #9, !6 # y#6.



Shading the Desired Regions as in the Text Shade to “Cross Out”

As you get the intersection of more regions, it gets harder and harder to identify the“multiple cross-hatching” of the region satisfying all of the inequalities if you shade asin the text. A suggestion is to reverse the shading which amounts to shading the part ofthe plane which you desire to “cross out.” This reverse shading leaves the commonintersection white. Then when you copy your result onto paper, shade only the “whitearea” to get a picture similar to Figure 9.31

Using the graphing style to “shade above” or “shade below” will only work forinequalities that can be solved for y. This is best, if we can do it, because we can trace onthe bounding curves for the region and find intersections to active function graphs. Inother situations, we use ½ Shade to shade between a lower function and an upperfunction over a perhaps more limited x-interval. The syntax for creating this drawn objectis Shade(lowerfunc,upperfunc[,Xleft,Xright,pattern,patres]) where the optional variablepattern is an integer 1-4 and patres is an integer 1-10.

Example 9.7.5 (p. 772)x y

x y

y

x

+ ≤− + ≤

≥ −≤

4

2 1

1

2Shaded to “Cross Out”



Example 19.7.7 (p. 776-7)

x y2 2

9 41− >

Shade Desired Region

Linear ProgrammingA TI-89/92/92 Plus can be great aid in identifying the feasible region, locating

vertices, and evaluating the objective function at the vertices. Here we demonstrate thisby working Example 9.8.1 (page 789-90).

Minimize 100 60subject to 250 +250 750

0.6 +0.06 0.7212 +60 60

0, 0

K x yx yx yx yx y

= +≥≥≥

≥ ≥

Note that we can handle the inequality x$ 0 by simply setting the viewing window so thatwe only see x-values which are positive (thereby avoiding a “Shade” command). Weshade to “cross out”, leaving the white area as the feasible region. Then we find a vertex,and we return to the home screen to evaluate the objective function using the coordinatesof the point.

Shade all below to “cross out”, 0 # x # 20, !4 # y # 15, Trace y2 and press x = 0

Chapter 10 - Integer Functions and Probability 56


Intersection of y2 and y-axis, Intersection of y1, y2 and then back to Home screen

Intersection of y1, y3 and then back to Home Screen, Zero of y3

Last vertex in objective function, scrolling up in history to find minimum

Chapter 10 - Integer Functions and Probability

SequencesThe TI-89/92/200 familyof calculators has a sequence graphing mode, where you can

enter either a formula for defining the sequence or a recursive definition for the sequence.The graphing variable in sequence graphing mode is n. To match the notation of the text,we start with nMin = 1 so the first term of our sequences will be u1(1) = a1. Note that youmust enter an initial term ui1 even when typing a formula. Once the sequence is definedin the Y= Editor, it can be plotted in the graphical screen, evaluated in the HOME screen,or investigated in a table. The trace and value commands are available in the sequencegraphical screen. First here is a sequence defined by the formula an = 4 + (n ! 1) * 6.



Consider Example 10.1.10 (page 815) an = 2an-1 + 5, a1 = 3.

We can also create a list of a finite number of terms in a sequence given by a formulausing the MATH menu, List submenu 1:seq( command. You use any variable name asthe index for the sequence in the formula, give the variable name, the start, and the end.



SeriesThe simplest way to compute a series is to use the MATH menu, List submenu

commands seq( and sum( together as on other TI calculators. Consider parts a. and b.of Example 10.2.2 (page 822).

21

5k

k =

∑ ii

2

1

6

=

∑

Even better is using the symbolic summation operation found in the Calc menu ….Many symbolic summations can be simplified, and some infinite series can be evaluated.

We can also investigate a finite series by entering u1(n) = an + 1 and u2(n)akk

n

=∑ 1

= u2(n ! 1) + u1(n ! 1) with u1(1) = a2, u2(1) = a1 in the Y= Editor using sequencegraphing mode.

Permutations, Combinations, Random NumbersMany questions in probability involve the use of factorials, permutations,

combinations, and experiments with random numbers generated by computer orcalculator. Commands for these operations can be found in the MATH menu, Probabilitysubmenu.



Use the built-in commands for nPr and nCr rather than the formulas involving factorialsbecause doing so allows n to be larger. For n $ 450 a floating-point factorialcomputation will overflow on a TI-89/92/200, but you can still compute furtherpermutations and combinations for these larger values of n.

ti-89/voyage 200/ti-92 online graphing calculator manual ... · ti-89/voyage 200/ti-92 online...

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