ti-83 plus/ti-83/ti-82 online graphing calculator manual for dwyer

TI-83 Plus/TI-83/TI-82ONLINE Graphing Calculator Manualfor Dwyer/Gruenwald’s

PRECALCULUSA CONTEMPORARY APPROACH

Dennis PenceWestern Michigan University

Brooks/ColeThomson Learning™

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Table of Contents

TI-83 Plus/TI-83/TI-82 Graphing Calculators

Chapter 1 Foundations and Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Calculator Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Order of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Exponents and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Fractional Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Function Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Graphing a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Rational Functions and Vertical Asymptotes . . . . . . . . . . . . . 17

Chapter 2 Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Increasing and Decreasing, Turning Points . . . . . . . . . . . . . . 20Combinations and Composition of Functions . . . . . . . . . . . . 20Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Graphing a Family of Functions . . . . . . . . . . . . . . . . . . . . . . 21Piecewise-defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . 22Least-Squares Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 3 Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 24Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 4 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . 27Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Regressions Involving Exponentials and Logarithms . . . . . . 28

Chapter 5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Sine, Cosine, and Tangent Function Keys . . . . . . . . . . . . . . . 31Plotting the Sine, Cosine, and Tangent Functions . . . . . . . . . 32Families of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 32Cosecant, Secant, and Cotangent Functions . . . . . . . . . . . . . 33Plotting the Inverses of Sine, Cosine, and Tangent . . . . . . . . 33

Chapter 6 Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . . 34Graphical Check of Equations . . . . . . . . . . . . . . . . . . . . . . . . 34Conditional Trigonometric Equations . . . . . . . . . . . . . . . . . . 35

Chapter 7 Applications of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Complex Numbers Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 36Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Plotting Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 8 Relations and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Graphing Relations in Pieces . . . . . . . . . . . . . . . . . . . . . . . . . 40Plotting Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Plotting Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Conics Flash Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Plotting Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 9 Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . 43Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Identity Matrices, the Inverse of a Matrix, Determinants . . . 46Systems of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 10 Integer Functions and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 50Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Permutations, Combinations, Random Numbers . . . . . . . . . . 52

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-83 Plus/TI-83/TI-82, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc.

TI-83 Plus/TI-83/TI-82The TI-83 is an excellent choice for a graphing calculator to use while learning from

Precalculus. The older TI-82 will do most of the activities presented here, but it isseriously lacking if you continue on to study statistics or the mathematics of finance later.Directly out of the box, the newer TI-83 Plus has exactly the same features as a TI-83(with the relocation of one key). However the TI-83 Plus has more memory and flashROM, enabling it to be electronically upgraded and to add further applications. Thenewest TI-83 Plus Silver Edition has a faster processor, even more memory, includesmany cost applications, and includes the GraphLink cable (actually quite a bargain). Youare encouraged to look at the Texas Instruments graphing calculator web pages(http://education.ti.com) to find the latest information on free or commercial TI-83 Plusapplications that can be downloaded using a computer and the GraphLink cable. Alsocheck for the newest operating system (OS) at that web site for the TI-83 Plus. A newerOS may fix problems and pave the way for newer applications. Thus the TI-83 Plus(Regular or Silver Edition) should be your choice if you are purchasing a new calculatorin this family.

Chapter 1 - Foundations and Fundamentals

Calculator FundamentalsWhen you turn on a TI-83 Plus, TI-83, or TI-82, it usually comes up in the Home

screen. If not (because the calculator did an “automatic shutoff” in another screen), pressy [QUIT] to move to the Home screen where immediate computations are performed.The ‘ key performs two important activities here. While you are typing a newcommand line (before Í), pressing ‘ will clear out everything in the commandline. If there is nothing in the command line, pressing ‘ will clear out all of theprevious results still showing in the Home screen.

Press z so that we can check (and explain) the various mode settings.

TI-82 MODE Screen TI-83 MODE Screen

http://education.ti.com

Chapter 1 - Foundations and Fundamentals 6


The first two lines determine how the calculator will display real numbers. Normal (thedefault) tries to show the entire number normally, but switches to scientific notation if apositive number is too large or too small. Sci always uses scientific notation, and Eng

uses a special scientific notation where exponents are a multiple of 3. Float (the default)moves the decimal point or the scientific exponent to show 10 significant digits (with zerosuppression to the right). If we select one of the digits 0123456789, the results aredisplayed rounded to that many decimal places. For now select the default setting on everyline (the left-most choice) by pressing cursor keys to highlight the desired selection andthen pressing Í. Briefly, the third line specifies the angle mode, the fourth-sixthlines set graphing, the seventh line (TI-83) sets the complex number format, and the lastline determines the split screen (if any).

The keyboard layout is fairly simple. Pressing a key does what is printed on the key.Pressing y (you do not need to hold it down) and then another key gives the operationprinted above, left, and in the same color. Pressing ƒ (you do not need to hold itdown) and then another key gives the operation printed above, right, and the same color(usually a letter). Many keys bring a menu to the screen, perhaps with further submenus.For example, the � key brings up the MATH menu, whereyou cursor right or left to change submenus (MATH, NUM,CPX, PRB) and cursor up or down to highlight a command.You select a command by highlighting it and pressing Í

or by just pressing the number in front of the displayedcommand. An arrow means there are more commands, eitherup or down. The TI-83 Plus/TI-83/TI-82 family of graphingcalculators does not allow you to type commands by typingcharacters one-by-one using the ƒ keys. The only alternative to finding a commandin a menu is to use the [CATALOG] where all commands are listed in alphabetical order.(Unfortunately no catalog is available on the TI-82.) For TI-83 Plus users, I would highlyrecommend installing the free flash application Catalog Help.

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

order of operation can be controlled with parentheses. Thiscalculator allows implied multiplication (no multiplicationsymbol is needed between the two objects) in many situationswhere there is no other interpretation. Just be careful withimplied multiplication, because if there is any otherinterpretation possible, something else will happen. Finalparentheses can be omitted. The TI-82/83/83 Plus family



assumes all “missing” right parentheses are needed at the end of the expression.It is very important to recognize the difference between the blue subtraction key ¹

above the Í key and the grey negation key Ì 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 the keyis pressed and one after. The negation operation takes onlyone number as an argument coming after the key is pressed.If you start a new command line by pressing the subtractionkey ¹, the calculator assumes you wish to do a continuationcalculation. Thus it assumes that you want to subtractsomething (yet to be typed) from the previous answer. Youcan also get the previous answer anywhere within thecommand line with y [ANS] which is found above the negation key.

There are many situations where you want to execute essentially the same commandrepeatedly. There are some nice editing features that makethis easy to do. The command y [ENTRY] found above theÍ key causes the last command line to be recalled so thatyou can edit it. Pressing y [ENTRY] several times allowsyou to go back to several previous command lines (limited bythe size of some memory buffer). When you edit a previouscommand line, you do not need to move the cursor point tothe end before pressing Í. If you want to execute exactly the same command line,you do not need to recall it. Just repeatedly press Í. In the screen shown here, wehave typed 11 Í and then pressed Ã 7 Í. As we repeatedly press Í, weadd 7 to the previous 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 →. Youfollow this command by a single letter (only capital letters are enabled). Then when youneed to use the result later, you simply type the letter (with the ƒ key). There is noway to “delete” one of these memory locations, but yousimply replace the old value with a new one when you storesomething new there. It saves time if you store intermediatecomputations rather than copying down a number and retypingit later. Further, most people are lazy, and they copy downonly a few of the decimal places. The “storing” operationsaves the complete number with all significant decimal placesfor later use.



Complex ArithmeticThe TI-83 and TI-83 Plus can handle complex arithmetic. Press z and select

a+b rather than Real. The symbol for the imaginary is a second function on thekeyboard above the decimal point. Do not try to use the letter I above the ¡ key.Typing a number immediately before is one place where you can safely assume impliedmultiplication. You can then add, subtract, multiply and divide complex numbers. In theMATH menu, the CPX submenu has other commands for complex numbers.

The absolute value function in the MATH menu, NUMsubmenu, has the traditional meaning for real numbers. Fora complex number, abs gives the modulus (or square root ofthe sum of the squares of the entries). In either case this resultrepresents the “length” or “size” of a number, and is alwayspositive (unless the number is zero).

Scientific NotationEven in our Normal mode, a number may be expressed in scientific notation if it is

too large. Calculators and computers have a short-hand for this. Instead of printing out5.7319 × 1025 which is difficult, they simply present 5.731925. You should use thesame short-hand when you want to enter a number in scientific notation (avoidingmultiplication by a power of 10). Use the y []where you want this symbol to beplaced. Internally the calculator uses this notation, and 9.99999999999 is the largestnumber it can handle. If a computation results in a larger number, there will be an errormessage. 1⁻99 is the smallest positive number represented, and positive numberssmaller than that are rounded to zero.



Exponents and RadicalsThere are special commands to square and cube a number. Squaring is the key ¡

and cubing is found in the MATH menu, MATH submenu 3:₃. Further the — key raisesa number to the negative one exponent. To raise a number to any exponent other than 2,3, or -1, use the › key. This command also works for negative and fractional exponents.

Similarly there are special commands for square root (above ¡) and cube root (in MATHmenu, MATH submenu). For other radicals, use the MATH menu, MATH submenucommand 5:x√ such as the sixth-root of 64 above.

Fractional ArithmeticAll of the calculators in the TI-82/83/83 Plus family are

numerical calculators. They do not strictly do any symbolicoperations such as fractional arithmetic. There is, however,a command that attempts to convert a numerical answer intosome “nearest fraction” that can be useful if you want tocompare your result to a simple fractional answer that mightbe given by someone working by hand. The command in theMATH menu, MATH submenu is 1:Frac .

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

Pt-On from the DRAW menu, POINTS submenu. Issuing this command from the Graphscreen, you get to select the point with the free-moving cursor (and Í). Issuing thiscommand from the Home screen, you type the desired coordinates. Either way, the



resulting point on the Graph screen is a drawn object that goes away if you resize theviewing window or regraph anything.

A more permanent way to plot several points is to use a statistical plot. Suppose wewish 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

To make sure that the statistical list editor is in the default configuration, press y

[CATALOG], then S, and select the command SetUpEditor. (This is not needed and notavailable on the TI-82.) Press Í to execute this command in the Home screen. Weenter this data by pressing … to bring up the STAT menu and selecting 1:Edit fromthe EDIT submenu.

Your lists displayed in this statistical list editor may or may not contain old data. Thequickest way to clear out old data here is to do the following. Cursor up to highlight thename of the list, say L1. Press Í to move the cursor down to the command line at thebottom of the screen. Press ‘ to empty out the command line, and then press Í

to make this “empty” list the definition of L1. Empty out L2 in the same way.



Type the desired x-values in list L1, and then type the corresponding y-values in list L2.It is easy to delete mistaken entries and to insert additional entries with the { and [INS]

keys. After the data has been correctly typed, press y [STAT PLOT] to specify thestatistical plot details. Select Plot1. Highlight and then select Í to match thefollowing screen. The first Type is a scatter plot and the first Mark is a box.

Before plotting, we need to set the viewing window and we need to make sure thatnothing else will appear in our graph. Press o and make sure that no function formulais selected (by having its “equal sign” highlighted). If one is highlighted, move the cursorto it and press Í to deselect that function formula. Press q to bring up somequick ways to reset the window. For example, 9:ZoomStat will always resize thewindow so that you can see all of the data in a statistical plot. Here we have other reasonsfor preferring 4:ZDecimal so that pixel coordinates come in even tenths. After gettingthe graph, we check to see what viewing window settings were fixed by pressing p.

In any graph, moving the cursor keys activates a free-moving cursor point in the plot.The coordinates of this free-moving cursor are displayed at the bottom of the screen.



Now you see why we like this “nice” viewing window. Pressing r activates somekind of tracing action in the plot. For a statistical plot, we can see the coordinates of thepoints in the scatterplot as we cursor right and left. Suppose the reader is asked toestimate the y-value when the x-value is 5.1. Since 5.1 is outside our viewing window,we need to resize the window. Change Xmin to 0 and Xmax to 9.4, and then press s

and a cursor key to get a free-moving cursor point to put in the approximate location.

Free-moving Point Trace Estimate at x = 5.1

Function GraphingMake sure that the graphing mode is Func in order to graph functions of the form y

= some expression which is then typed in the Y= screen. Alsolet’s make sure that all of our function plots look the same byselecting the same formatting options on the y

[FORMAT] screen, matching the one on the right here. Forexample, let’s graph the function y = 3 x2

! 12 x + 14 inthe standard viewing window, as demonstrated in page 54-55of the text. Clear out any other functions that may be storedthere, and make sure that no statistical plot is highlighted (meaning it is turned on). Toturn off a statistical plot, move up to the highlighted plot number and press Í tochange. Type the formula in slot Y

1, press q, and select 6:ZStandard as shown.

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 little



more detail. The zoom command 0:ZFit resets Ymin and Ymax so that the graph justfits with the screen for !10 # x # 10. Notice that we cannot see the x-axis any longerbecause the setting for Ymin is positive.

Press q, cursor right to MEMORY, and select 1:ZPrevious to get back to the graphbefore the last zoom operation. Then try a q, 1:ZBox , selecting a box around theparabola that nicely includes some of the axes for yet another view.

There are many nice operations that can be performed while looking at a graph. Ther turns on a blinking pixel that can be moved right or left along the curve, showingthe coordinates at the bottom of the screen. The x-coordinates are pixel coordinates justas with the free-moving cursor, but the y-coordinates are actual function evaluations.Although we do not need it here, there are two nice ways to change the viewing windowwhile tracing. If you press Í while tracing, the window will shift so that the blinkingpixel being traced moves to the center of the viewing window (called a Quick Zoom). Ifyou trace all the way to the left or right edge of the graph and then continue to try to gofarther, the window will shift to let you continue (called panning).

Pressing [CALC] and selecting 3:minimum allows the estimation of the minimum ofthe function on a subinterval. You input a lower bound and a upper bound to define thesubinterval and help the routine with a guess (usually by moving the cursor point nearwhere there is an apparent minimum).



Consider y = 0.018 x4! 0.45 x3 + 2.93 x2

! 1.5 x + 61.5 for0 # x # 12 . We get the following plot.

Tracing does not give integer x-values as we might want but the [CALC] command1:value allows us to evaluate the function exactly at specific x-values such as integers.While tracing, you can also type the exact x-value desired instead of moving with thecursor keys. Below we trace to an apparent maximum, use value to find the largest valueat an integer, and use the [CALC] command 4:maximum to explore this function.

By Trace By Value at Integer 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 of solvingfor the x-intercepts and y-intercepts for the function y = 1000 x3

! 15 x2 + 0.0002from Example 1.5.8 (page 73). We type the formula in the o screen and begin in thestandard viewing window with q 6:ZStandard as suggested in the text. Then we useq 1:ZBox several times to narrow in to a more appropriate viewing window as



indicated below.

ZStandard 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 [CALC] command 2:zero tobegin a numerical routine to solve for the zero or root of this function. The routine asksthe user to give a left bound and a right bound to specify the subinterval where you desireto know the root. It is easy to give these bounds by moving the cursor point a little to theleft of the apparent zero on the graph and pressing Í. Then move the cursor pointa little to the right of the apparent zero on the graph and press Í.

Then provide an initial guess for the zero, again by moving the cursor point to very nearthe apparent zero on the graph. The numerical routine works more rapidly if a moreaccurate initial guess is given. Repeat this procedure, giving different subintervals, to findthe remaining roots. Finally [CALC] 1:value followed by 0 displays the y-intercept.



Remember that this procedure will only locate an intercept contained within your viewingwindow. The user might need to look at other larger viewing windows to be confidentthat this function has no other intercepts outside the ones wehave considered. Panning and quick zooms might also help.

There is also a [CALC] command 5:intersect tonumerically find an intersection point for the graphs of twofunctions. Consider the two functions of Example 1.5.9 (page75), y = x3

! 7 x2 and y = 14 ! 17 x, in the viewingwindow with !2 # x # 8 and !60 # y # 30 . This commandprompts for the user to confirm the desired two curves and to specify an initial guess tostart its numerical routine.

Finally in the � menu, MATH submenu, 0:Solver... on the TI-83 and 0:solve(

on the TI-82, there are ways to solve equations in the Home screen. For example, the x-value of the intersection point above is simply a solution to the equation0 = x3

! 7 x2! 14 + 17 x. Again you can speed the routine by giving an initial guess

for x, and you can specify a subinterval to limit the search with the line labeled bound.You begin the routine after things are set by highlighting the desired variable andpressing ƒ [SOLVE].



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 in the y-direction. The command q 5:ZSquare will always change the viewing window toone 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 and first in the standardy x= −64 2 y x= − −64 2

window and then after q 5:ZSquare. There is also a [DRAW] 9:Circle( commandto draw a circle, but drawn objects like this cannot be traced.

ZStandard Then ZSquare Circle(0,0.8)

Rational Function and Vertical AsymptotesThus far, we have been using the mode setting CONNECTED to get nice graphs of the

smooth functions considered. The calculator does this by plotting points (which are theones you see when you trace), and then by turning on other pixels in the plot to make itlook like those points are connected by short line segments. Most calculator andcomputer plots work this way by default. For rational functions, this connecting of thedots leads to a deceptive picture. It is better to convert to the DOT mode (or to at least look

at both). Consider in the standard viewing window. Notice thatyx x

=+

+−

−18

2

2

35

the near vertical lines at x = !2 and x = 3 appearing in the connected mode (where thisfunction has vertical asymptotes) do not appear in the dot mode.

Chapter 2 - Functions and Their Graphs 18


Connected Mode Dot Mode

Chapter 2 - Functions and Their Graphs

Evaluating FunctionsAfter a function formula has been stored in the o editor, there are several ways to

calculate and display the value of the function. The simplest is to calculate function

values in the Home screen. Consider fromP v v v( ) .= +0 0178678 2.011683

Example 2.1.14 (page 130). We store this in Y1 using the graphing variable X, and thenget Y1 from the � menu, Y-VARS submenu, 1:Function sub-submenu. Noticefunction notation will take precedence over implied multiplication.

The Solver allows us to quickly answer the question as to what velocity gives a poweroutput of 500,000,000 watts. We can also get these results while looking at the graphicalscreen using the trace and value commands. Note that you can actually type 20 while



you are tracing to get the exact evaluation at x = 20. If we alsoplot Y2 = 500000000, then we can seek the intersectionbetween the two graphs. Here the viewing windows are all 0# x # 3500, 0 # y # 600,000,000.

We can also look at a table of values. In the Table Setup [TBLSET], we can choosebetween having the table entries automatically generated using the TblStart and �Tbl

values or having the table entries determined by asking the user.

We can even look at the graph and a table at the same time using the split screen or G-Tgraphing mode. On a TI-83, if you start the trace operation, the table changes to matchthe pixel and evaluation coordinates showing at the bottom of the graph as you trace.



Now is a good time to mention the best way to choose a viewing window for a plot of anew function. First put the formula for the function in the o editor. Then press TableSetup [TBLSET] and set the TblStart and �Tbl values so that we will get a table offunction values where we think we want the interval [Xmin, Xmax]. The third step isto press [TABLE] to look at the function values. As we scroll through these functionevaluations, take note of how we will need to set [Ymin, Ymax] if we stay with theoriginal idea about [Xmin, Xmax]. Often we will decide to change even the x-intervalas well after looking at a table of the function values. The fourth step is to set p

based upon what we observed in the table. Finally press s to see a plot that at leastincludes 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 [CALC] commands

3:minimum and 4:maximum while viewing the graph. Consider f x x x( ) = − +18

3 2 2

from Example 2.2.8 (page 149). The graphs below are in the standard viewing window.

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

with combinations and compositions without retyping, both in Home screen and in furtherfunction slots in the o editor. We can only plot or evaluate these. There is nosymbolical operation to simplify the new functions created by these operations. Considerf(x) = 2 x2 + 4 x + 5 and g(x) = 2 x + 1 from Example 2.3.4 (page 163).



Inverse FunctionsThe commands [DRAW] 6:DrawF and 8:DrawInv plot a non-interactive graph of a

function and the inverse of a function. Notice that this command for the inverse is reallyjust interchanging the x-coordinates and y-coordinates for plotting purposes. Thefunction does not need to be one-to-one and may not have a true functional inverse. Stillthe plot is correct when the function has an inverse.

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 can usually be handled on a TI-83/82 using logical tests.

The [TEST] menu provides the various inequality and equality symbols. A logical teston a TI-83/82 evaluates to 1 if it is true and 0 if it is false. We use this to “zero out” partsof a formula when we do not want that part to contribute. Consider

f xx x x x

x x( )

,

,=

− + − + <

− ≥

RST

3 26 9 4 3

3 3

from Example 2.5.5 (page 187). Notice that in the connected mode, the nearly verticalline between dots connects the two pieces where it is not appropriate. The dot mode doesnot do this (although it also leaves dots within pieces unconnected as well).

Connected Mode or Style Dot Mode or Style

The difficulty with this way of representing piecewise-defined functions is that all of thepieces must be defined for all numbers x in the x- interval to be considered, even whenyou might not be using that piece at that x-value. For example, the formula



Y1 = (x+8)(x<−4)+(√(16-x2))(X≥−4 and x≤4)+(x-8)(x>4)

will only be defined for !4 # x # 4 and will give an error message (or not plot) outsideof this subinterval because the middle piece is not defined there. A “fix” for thisparticular example is the following.

Y2 = (x+8)(x<−4)+(√(abs(16-x2)))(X≥−4 and x≤4)+(x-8)(x>4)

Least-Squares Best FitThe TI-83/82 provides several different regression fits for numerical data, including

using linear, quadratic, cubic, and quartic polynomials. We demonstrate here onlya linearfit. Consider Table 2.10 (page 208) giving U.S. health-care expenditures (in billions ofdollars) 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). Enterthe years in list L1 and the expenditures in list L2 in the … editor. As we did earlierin this chapter, turn on a statistical scatter plot of this data and use q ZOOM 9:ZStat

to size the viewing window in an appropriate manner for this data.

To see the regression coefficient r, go to the [CATALOG] and select DiagnosticOn. Press… CALC 4:LinReg(ax+b) to have this regression performed. The optionalarguments after the regression command specify the two lists and indicate the function

Chapter 3 - Polynomial and Rational Functions 24


slot where we want the formula to be stored. After this computation, the coefficients aand b and the formula for the regression equation can be found in � 5:Statistics

EQ to be used later.

If you give no arguments, the command LinReg assumes data will be in lists L1 and L2.In a statistics course you will learn to interpret the significance of the diagnosticcoefficient r. We will simply note that when r 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.

Y1(2003) = 59.118 (2003) !116947.69 = E(18) = 59.118 (18) + 401.54 = 1465.664

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.



Here is one trick for making the evaluation of a high degree polynomial more accurateand the graphing of it more rapid. Algebraically we can rewrite a polynomial in severalequivalent ways. For example,

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 the first,we can avoid using the “power key”› which is quite slow forrepeated computations, and we reduce the total number ofarithmetic operations required in evaluation. On virtually allgraphing calculators, entering a fifth degree polynomial in thesecond way will cause it to plot in about half the time asentering it the first way.

Polynomials can get verybig, making theq ZOOM 0:ZFit a veryattractive optionafter you have set the x-interval for the desired window. We use this to plot the abovefifth degree polynomial. (Usually you will want to go back to the window screen andreadjust the Yscl as was done below after the window is set by ZFit.)

Then to study end behavior, zeros, and turning points you will probably want to zoom outto check end behavior and zoom in to better see zeros and turning points.



A TI-83/82 graphing calculator has no special features for symbolic operations withpolynomial multiplication, division, or complex roots. For the TI-83 Plus, there is a freeflash application called PolySmlt that can find the roots of polynomials (real andcomplex). Here we demonstrate with two different polynomials. First we consider

Example 3.3.1 (page 245) .4 3 2( ) 5 8 29 20 12f x x x x x= + − − +

Next we consider Example 3.3.6 (page 251) .5 4 3 2( ) 2 4 2 4g x x x x x x= − − + − −

Chapter 4 - Exponential and Logarithmic Functions 27


Note that any numerical root finding algorithm will have trouble with a double root. Herethe polynomial g(x) has !1 as a double root. This is approximated by the two complexroots , each with very small imaginary part. This is not aE1 3.050442591 -7i− ±mistake, but simply the result of the fact that when we round in numerical computations,we effectively get the roots of a slightly different polynomial.

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

mode generally is best for this familyof 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.



Two special exponential functions are provided on the keyboard, 10x and ex, and youshould use these rather than the power key for more accuracy. This special number eappears as a y keystroke above ¥. In addition, you can use the natural exponentialkeystroke to get the value of this number e with e^(1), giving 2.718281828. Usingthese methods will be better than typing these digits because even the guard digits youcannot see will be correct.

Logarithmic FunctionsThe two special logarithmic functions provided on the keyboard give common

logarithms, «, and natural logarithms, µ. Use the Change-of Base Formula (page326) to work with logarithms in another base in terms of one of these special ones.

loglog

log

ln

ln, ,

au

u

a

u

aa u= = ≠ >1 0

Regressions Involving Exponentials and LogarithmsThe … CALC menu offers a number of regression options that involve families of

exponential and logarithmic functions. The preliminarysteps for these regressions are thesame as for linear regression above in Chapter 3. You simply select and plot a differentregression fit. You can even plot several on the same screen and decide visually whichseems to be the best fit.

LnReg a + b ln(x)ExpReg a bx



PwrReg a xb

Logistic (only on TI-83/83 Plus)c

a e b x1 + −

For example, consider the data from Table 4.7 (page 345 describing a state deerpopulation (in thousands) since 1999 (t = 0). We show how to obtain an exponential fitfor the data.

Year (since 1999) 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 two lists and obtain a scatter plot. Then perform the exponentialregression, and save the regression equation in a function slot. Finally, compare thescatter plot to the graph of the regression equation.

If we simply evaluate this at X = 6, we get the prediction Y1(6) = 23017.5674. We canalso follow the instructions in the “Calculator Keys” box on page 346 to convert the baseof the exponential function given by the calculator to the natural base e. Remember thatimmediately after doing a regression, the coefficients (here a and b) can be found in�5:Statistics EQ so that they do not need to be retyped in the home screen.

Chapter 5 - Trigonometric Functions 30


( ) ( )( )( ) ( )

ln 1.149206874

0.13907202960.1390720296

( ) 9992.407418 1.149206874 9992.407418

9992.407418 9992.407418

tt

t t

P t e

e e

= =

= =

The logistic regression is a very difficult computation. The routine in the TI-83 mayfail to converge. It seems to have problems with large data. In particular, it did not seemto work for this data set.

Chapter 5 - Trigonometric Functions

Angle MeasurementThe TI-83/82 has an angle mode setting of either Radian or Degree in the mode

screen. We will experiment here with both settings. Pressing y [ANGLE] brings up amenu with further angle commands. The first 1:° causes the number before this symbolto be interpreted as degrees, regardless of the angle mode. The second 2:' gives minutesand the third 3: gives radians, again regardless of the angle mode. The double quotesymbol ƒ ["], found above Ã, also serves as the notation for seconds. Note thatthere is also a special keystroke for Ä above the › key.

Assuming degree mode setting, expressions given in degrees-minutes-seconds (DMSnotation) will be converted to decimal degrees. The command y [ANGLE] 4: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 degrees-minutes-seconds (DMSnotation) will still be converted to decimal degrees (but with no indication to interpretanswer in degrees). The command y [ANGLE] 4:DMS still converts something indecimal into DMS, interpreting the decimal as decimal dgrees. Expressions designated



with only the degree symbol °will be converted into radians. In radian mode, the radiansymbol does nothing.

Sine, Cosine, and Tangent Function KeysThe keys ˜ ™ š interpret their argument based upon the angle mode unless

a degree or radian symbol is present to override the angle mode. Note that being able tooverride the angle mode should mean that you do not need to change a mode setting toswitch back and forth between degrees and radians for simple trigonometriccomputations. The most common error made when working with these functions is to bein the wrong angle mode. A goal should be to know enough about trigonometricfunctions so that you can immediately recognize when you start to get answers appropriatefor the wrong angle mode. Note that the trigonometric keystrokes on a TI-83 /83 Pluscome with a left parenthesis (and expect you to either type the right parenthesis or itassumes one at the end of the line). On the TI-82, no parenthesis is automaticallyprovided and the order of operation may give you surprising results (i.e. trigonometricoperations take precedent over multiplication and division).

Degree Mode, TI-83 Radian Mode, TI-83 Radian Mode, TI-82

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.



Degree Mode, TI-83 Radian Mode, TI-83 Radian Mode, TI-82

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 q ZOOM 7:ZTrig resets the viewing window to

Degree Mode !352.5 # x # 352.5, Xscl = 90, !4 # y # 4, Yscl = 1.

Radian Mode− ≤ ≤ =RST − ≤ ≤ =

6 152285613 6 152285613

4 4 1

2. .

, .

, ,x Xscl

y Yscl

π

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

Sine Connected Cosine, Tangent Dot Cosine, Tangent

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 L1 ={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).



Cosecant, Secant, and Cotangent FunctionsThere is no keystroke for the remaining trigonometric functions on the TI-83/82. You

need to know the fundamental identities for how csc x, sec x, and cot x are related to sinx, cos x, and tan x (namely that they are reciprocals).

For csc x type sin(X) or 1/sin(X).

For sec x type cos(X) or 1/cos(X).

For cot x type tan(X) or 1/tan(X).

We will leave as a challenging exercise the task of determining what to do for the inversesof the cosecant, secant, and cotangent functions. You are well advised, however, to avoidthe need for these by converting your task into a question about the inverse of the sine,cosine, or tangent.

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 q

ZOOM 7:ZTrig still gives a reasonable viewing window, although we may only be usinga small 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.

Chapter 6 - Trigonometric Identities and Equations 34


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.

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

For potential identities, it is actually more convincing to look at tables of the twoexpressions evaluated for the same x-values. The graphs may look the same but merelybe close. The table entries appear to be exactly the same.

Example 6.1.2 (page 463)continued

Chapter 6 - Trigonometric Identities and Equations 35


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

Here the graphs which clearly never intersect will be more convincing than a table.

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 guess 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 manageto reduce the problem to something such as

,cos x =−1 3

2then we can use [COS

-1] and our knowledge about the

reference angles to solve for x in the interval [0. 2B).

Chapter 7 - Applications of Trigonometry 36


Chapter 7 - Applications of Trigonometry

Complex Numbers RevisitedRecall that the TI-82 has no complex number operations.The rectangular form for representing a complex number is a + b i, and there is a

mode setting to enable this on a TI-83/83 Plus. You can find the symbol i as a y

keystroke above the decimal point. The trigonometric form for representing a complexnumber is r (cos 2 + i sin 2) (page 538). This is available too on a TI-83/83 Plus in aslightly different notation called the polar form r ei2 . The variables r and 2 have thesame meaning in the trigonometric and polar forms. In fact, the definition of a complexexponential e" + i $ = e" (cos $ + i sin $) quickly reduces to ei 2 = cos 2 + i sin 2 .(See Exercise 7.4.51 on page 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. Here we assume radian mode.

In the polar complex number mode, the command � CPX 6:åRect converts to therectangular complex form. In the rectangular complex number mode, the command�CPX 7:åPolar converts to the polar complex form. Note that you can type in complexnumbers in any form. Often the resulting complex number is too long to see all of it onthe screen at once. Just press the right and left cursors to see the result before beginningto type the next command line. The modulus is obtained by the command abs.



The square root command and the power command (to getother nth roots) give principal roots (not all nth roots). Forexample, 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 argument 2.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 run exactly aswritten there on a TI-82. The only slight change that occurs on a TI-83/83 Plus is that thetrigonometric functions come with parentheses. While the old-fashioned commandIS>(K,N) indicating to “Increase K by 1 but Skip the next command if K > N” is stillthere on the newer calculators, there are now better ways to loop.

Executing NGON with n = 8 in window 1.516 1.516, 1 1x y− ≤ ≤ − ≤ ≤

Polar CoordinatesOn a TI-83/82, conversions between rectangular coordinates and polar coordinates

are implemented as commands in the [ANGLE] menu. The calculator has chosen toconvert the rectangular (0, 0) to the polar (0; 0). It gets a unique polar representation forrectangular coordinates other than the origin by selecting r > 0, 0 # 2 < 2 B.

Press [FORMAT], above the q key, and you will find the first formatting option



is to select rectangular graphing coordinates RectGC or polar graphing coordinatesPolarGC. This format option will determine the coordinates that appear at the bottom ofthe graphical screen in all graphing modes.

Plotting Polar EquationsOn thez screen, move from function graphing to polar graphing byselectingPol.

On the [FORMAT] screen, select polar graphing coordinates PolarGC. Then press o tosee the polar equation editing screen. Type in the formulas for

r = 2 cos 2 and r = 1 + 2 sin 2 .

from Example 7.5.7 (page 561). The graphing variable is now 2 so it can be obtainedeither by pressing „ or getting it as an alpha character (above Â).

In addition to setting the x-range and y-range on the p screen, 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 q ZOOM

4:ZDecimal to get a decimal, equally scaled viewing window and also to get [0, 2B] for2, with 2step = B/24 so that we hit favorite multiples of B as we trace.

The [CALC]menu no longer contains an “intersect” command, and there is a good reasonfor this. An apparent point of intersection of two polar equations can occur because ofone representation of that point in one equation and a different representation of that samepoint in the other equation. For example, the two polar equations plotted above appear



to have three points of intersection. By tracing to find the approximate polar coordinatesgiving the point on each equation and by turning on rectangular graphing coordinates aswell, we can roughly compute the following table describing these three points and howthey 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 for these intersection points on a TI-83 with the interactiveSolver, using this initial graphical work for starting guesses and for setting the equationsto be solved. Note that we can find the symbol for our polar equations in the�menu.

VectorsThere is no special vector data type nor are there any special vector operations on a

TI-83/82. The best that we can do is to store the components of a vector in either a list,a 1×2 matrix, or a 2×1 matrix. Any of these ways allows vector addition, vectorsubtraction, and the multiplication of a vector by a scalar to be computed.

Chapter 8 - Relations and Conic Sections 40


We would need to write short programs to implement the operations for finding the normof a vector or for finding a unit vector in the same direction as a given vector. Betteradvice would be to get a different calculator if you later find yourself in a course thatrequires significant computations involving vectors.

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 y [DRAW] 2:Line( and it expects as argument thecoordinates of the starting point and ending point. Unfortunately there is no simple wayto put an arrow at the end 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 vectorPQ

at the origin. Then we add another line segment for putting the initial point at thePQ

terminal point for P(-1, -2) using the command Line(⁻1,⁻2,⁻3,1). Our viewingwindow is from ZDecimal and we have RectGC as a format so that the free-movingcursor can help us locate endpoints.

Chapter 8 - Relations and Conic Sections

Graphing Relations in PiecesThere is no simple way to graph a general relation. To plot, we must solve the

equation for y (possibly with more than one solution or piece). Looking at Example 8.1.9



(page 600), we solve Just to highlight4 9 36 36 4 92 2 2x y y x+ = = ± −as b g / .

some potential difficulties, we select the window with ZTrig and plot both the upper (+)and lower (-) parts of the ellipse as separate functions (in function graph mode, RectGC).

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 = 2.8797933.The next pixel to the right has coordinate x = 3.010693, andin this column of pixels there is no plot. We do not landexactly on x = 3 as a pixel coordinate, where both Y1 and Y2

would evaluate to zero. Using ZDecimal on the right heredoes give pixel coordinates that include the integers as well asother exact decimal values.

Plotting ParabolasA parabola that opens upward or downward is easily plotted as a single function

because we can solve uniquely for y in the equation. For a parabola that opens right orleft instead, we can either plot two separate pieces (where we can trace on each piece) orwe 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 standard viewingx y+ = − −( )1 21

2

2

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 Example 8.4.3 (page 646).

Conics Flash ApplicaitonFor the TI-83 Plus, the flash application Conics gives very nice ways to explore all of theconic sections. Here we demonstrate with an ellipse.

Plotting Parametric EquationsA TI-83/82 calculator 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 „ key gives the graphing variable t. For the viewing windowbelow, we started first with the standard viewing window, and then did a ZSquare to getequally scaled axes.

Chapter 9 - Systems of Equations and Inequalities 43


Notice that when we trace, we can see the value of theparameter t as well as the x- and y-coordinates of the pointhighlighted. Pressing the right cursor key increases the valueof the parameter t (which will not necessarily cause the pointto move right). While you are tracing, you can also type adesired t-value. The window settings have changed forparametric equations as well. We set the t-interval for theparameter as well as the bounds for the axes. The setting Tstep determines the plottedpoints (which can then be traced). In the connected graphing mode, small line segmentsare drawn between the plotted (traceable) points. If Tstep is too large, these line segmentsmay not be small, and our plot may be rather crude. If Tstep is too small, it will take along time to plot the parametric equations.

Chapter 9 - Systems of Equations and Inequalities

MatricesA TI-82 can store five different matrices, and a TI-83/83 Plus can store ten. While

you can enter very small matrices in the home screen, it is more convenient to use thematrix editor. A key labeled � can be found on a TI-82 and TI-83. To make roomfor the blueO key (where they can place any additional applications that are load intothe TI-83 Plus flash ROM), they needed to move the matrix menu key. You will find y

[MATRX] above the — key on a TI-83 Plus. This is the only keyboard change betweena TI-83 and a TI-83 Plus. You must always get the name of a matrix from this �menu. Matrix names will appear on the screen as a letter surrounded by square brackets,but you cannot type a left bracket, an ƒ character, and a right bracket, one characterat a time, in the home screen and have it mean a matrix name. Below we create threematrices and show how to do simple matrix arithmetic.



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. Generally when you execute one

of the elementary row operations, you will want to store the result in some matrix slot.



If you want, you can store successive results in the same matrix, overwriting the previousinformation as we do 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 completeGaussian elimination process on a matrix (not on the TI-82).The command is ref( to convert to a row-echelon formequivalent to the starting matrix. Gauss-Jordan elimination isdone by the command rref( to convert to the unique reducedrow-echelon form equivalent to the starting matrix. If theresult is too large to view at once, scroll right or left to see allof it before beginning the next command line.



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 5:identity( by simply giving the sizedesired for this new square matrix. For a square matrix which has an inverse, the key—gives the inverse.

That gives us two ways to solve a system of linear equations such as2 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.

For the TI-83 Plus, there is a free flash application called PolySmlt that can nicely solvesystems of linear equations (including ones with many solutions).



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. These function graphing styles are only available on a TI-83/83 Plus. On a TI-82,the best you can do with traceable function plots is to plot just the lines.

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 common



intersection 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 y [DRAW] 7:Shade( to shade between a lower function andan upper function over a perhaps more limited x-interval. (This Shade command is foundon the TI-82.) The syntax for creating this drawn object is

Shade(lowerfunc,upperfunc[,Xleft,Xright,pattern,patres])

the optional variable pattern is an integer 1-4 and patres is an integer 1-6.

Example 9.7.5 (p. 772)x y

x y

y

x

+ ≤− + ≤

≥ −≤

4

2 1

1

2Shade to “Cross Out”

Example 9.7.7 (p. 776)

x y2 2

9 41− >

Shade Desired Region



Linear ProgrammingA TI-82/83/83 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).

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 last two inequalities (x $ 0, y $ 0) by simply setting theviewing window so that we only see x- and y-values which are positive. We shade to“cross out”, leaving the white area as the feasible region. Then we find an intersectionpoint using the intersection command, and we return to the home screen to evaluatethe objective function using the coordinates of the intersection point.

Chapter 10 - Integer Functions and Probability 50


Chapter 10 - Integer Functions and Probability

SequencesThe TI-82/83/83 Plus family of calculators has a sequence graphing mode, where you

can enter either a formula for defining the sequence or a recursive definition for thesequence. A TI-82 allows only a one-term recursion formula (u(n) in terms of only u(n-1)), while a TI-83/83 Plus allows both one-term and two-term recursion (u(n) in terms ofu(n-1) and u(n-2)). First we show how this works on the TI-83, and later we give anexample for the TI-82.

The graphing variable in sequence graphing mode is n which is now given by the key„, and this is the only place you can get this variable (since the alpha character Ndoes not work here). To match the notation of the text, we start with nMin = 1 so the firstterm of our sequences will be u(1) = a1. Note that you must enter an initial term u(nMin)even when typing a formula. Once the sequence is defined in the o screen, it can beplotted in the graphical screen, evaluated in the home screen, or investigated in a table.The trace and value commands are available in the sequence graphical screen. When youwish the type u, v, or w to refer to a defined sequence, these can be found as y

keystrokes above the number keys ¬−®.



Consider Example 10.1.11 (page 815) an = 2an-1 + 5, a1 = 3 done on a TI-82. Points ina sequence graph are represented by individual pixels which are hard to see. Thus we usethe connected graphing mode, where the individual points in the plot are connected byline segments, to better see the graph.

TI-82 Sequence Graphing Mode

We can also create a list of a finite number of terms in a sequence given by a formulausing the [LIST] OPS 5:seq( command. You use any of the alpha characters as theindex for the sequence in the formula, give the alpha character, the start, and the end.

SeriesThe simplest way to compute a series is to use the [LIST]

commands OPS 5:seq( and MATH 5:sum( together.Consider parts a. and b. of Example 10.2.2 (page 822).

21

5k

k =

∑ ii

2

1

6

=

∑



We can also investigate a finite series by entering u(n) = an and v(n) = v(n-akk

n

=∑ 1

1) + u(n) with u(1) = v(1) = a1 in the o screen.

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 � PRB menu.

Use the built-in commands for nPr and nCr rather than the formulas involving factorialsbecause doing so allows n to be larger. For n $ 70 the factorial computation willoverflow on a TI-83/82 but you can still compute further permutations and combinations.

ti-83 plus/ti-83/ti-82 online graphing calculator manual for dwyer

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