# TI-86/TI-85 ONLINE Graphing Calculator Manual for

TI-86/TI-85 ONLINE Graphing Calculator Manual for Dwyer/Gruenwald’s PRECALCULUS A CONTEMPORARY APPROACH Dennis Pence Western Michigan University Brooks/Cole Thomson Learning™ Australia • Canada • Mexico • Singapore Spain • United Kingdom • United States COPYRIGHT © 2004 by Brooks/Cole A division of Thomson Learning The Thomson Learning logo is a trademark used herein under license. For more information, contact: BROOKS/COLE 511 Forest Lodge Road Pacific Grove, CA 93950 USA http://www.brookscole.com For permission to use material from this work, contact us by web: http://www.thomsonrights.com fax: 1-800-730-2215 phone: 1-800-730-2214 Table of Contents TI-86/TI-85 Graphing Calculators Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Foundations and Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Calculator Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Order of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exponents and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Fractional Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Function Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Graphing a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Rational Functions and Vertical Asymptotes . . . . . . . . . . . . . 17 Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Evaluating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Increasing and Decreasing, Turning Points . . . . . . . . . . . . . . 20 Combinations and Composition of Functions . . . . . . . . . . . . 20 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Graphing a Family of Functions . . . . . . . . . . . . . . . . . . . . . . 21 Piecewise-defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . 21 Least-Squares Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 24 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . 26 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Regressions Involving Exponentials and Logarithms . . . . . . 27 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Sine, Cosine, and Tangent Function Keys . . . . . . . . . . . . . . . 29 Plotting the Sine, Cosine, and Tangent Functions . . . . . . . . . 30 Families of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 31 Cosecant, Secant, and Cotangent Functions . . . . . . . . . . . . . 31 Plotting the Inverses of Sine, Cosine, and Tangent . . . . . . . . 31 Chapter 6 Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . . Graphical Check of Equations . . . . . . . . . . . . . . . . . . . . . . . . Conditional Trigonometric Equations . . . . . . . . . . . . . . . . . . Chapter 7 Applications of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Numbers Revisited . . . . . . . . . . . . . . . . . . . . . . . . . Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plotting Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8 Relations and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphing Relations in Pieces . . . . . . . . . . . . . . . . . . . . . . . . . Plotting Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plotting Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plotting Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . Chapter 9 Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identity Matrices, the Inverse of a Matrix, Determinants . . . Systems of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 10 Integer Functions and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations, Combinations, Random Numbers . . . . . . . . . . 32 32 33 34 34 35 35 37 38 38 39 39 40 41 41 41 42 44 45 46 46 48 48 Note that in Acrobat Reader, each chapter and section in this table of contents is linked to the appropriate location in the document. Similarly, chapter and section titles in the document are linked back to this table of contents. Web links are also active if your computer has an internet connection. TI-86, TI-85 The TI-86 is a good choice for a graphing calculator to use while learning from Precalculus. The older TI-85 will do most of the activities presented here, but it lacks some editing and statistical features of the newer model. The biggest improvements in the TI-86 are the increased memory and the ability to load and run assembly language programs. You can look at the Texas Instruments graphing calculator web pages (http://education.ti.com) to find some programs that can be downloaded using a computer and the GraphLink cable. Thus the TI-86 should be your choice if you are purchasing a new calculator in this family. However Texas Instruments announced several years ago the intention to stop making this line. The TI-86 seems to be heavily discounted now (probably to sell out the remaining stock). If you are buying a new calculator, I would suggest seriously looking at other TI models with flash ROM which can be upgraded. Chapter 1 - Foundations and Fundamentals Calculator Fundamentals When you turn on a TI-86 or TI-85, it usually comes up in the Home screen. If not (because the calculator did an “automatic shutoff” in another screen), press y k to move to the Home screen where immediate computations are performed. The ‘ key performs two important activities here. While you are typing a new command line (before Í), pressing ‘ will clear out everything in the command line. If there is nothing in the command line, pressing ‘ will clear out all of the previous results still showing in the Home screen. Press y l so that we can check (and explain) the various mode settings. TI-85/86 MODE Screen The first two lines determine how the calculator will display real numbers. Normal (the default) tries to show the entire number normally, but switches to scientific notation if a positive 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) TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 6 moves the decimal point or the scientific exponent to show 10 significant digits (with zero suppression to the right). If we select one of the digits 0123456789, the results are displayed rounded to that many decimal places. For now select the default setting on every line (the left-most choice) by pressing cursor keys to highlight the desired selection and then pressing Í. Briefly, the third line specifies the angle mode, the fourth and seventh lines set ways to report vector components, the fifth line sets the graphing mode, the sixth line fixes the number base, and the last line determines the way to compute the derivative for certain calculus operations. 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 operation printed above, left, and in the color orange. Pressing ƒ (you do not need to hold it down) and then another key gives the operation printed above, right, and the color blue (usually a capital letter). Pressing ƒ twice locks you into this alpha setting so that you can type several letters at once. (Another press or an a will release you from alpha setting.) Pressing y ƒ gives lower case letters. Many keys bring a menu to the bottom line of the screen, perhaps with further submenus. The function keys %&'() are used to select the items in the bottom line of the screen. If there is a row of menu items in the next line up from the bottom, first press , and then one of the function keys to get that item via defgh. For example, pressing ,‹ brings up the MATH menu. Items in all capitols generally are submenus. Pressing % moves those submenu items up and gives a new bottom row of menu selections. Pressing & now pastes the command iPart to find the integer part of a number. Notice that the menus do not go away after you make a selection. Press - to drop the lowest menu (or press anything that brings up a new menu). An arrow means there are more commands to the right. Press . to see these additional commands in the lowest line. The TI85/86 family of graphing calculators also allows you to type commands by typing characters one-by-one using the ƒ keys, but this is rather slow. Since the correct spelling and spacing come from the menu, this will be the preferred way. Another alternative to finding a command in a menu is to use ,[CATALOG] on the TI-85 or ,v% on the TI-86 where all commands are listed in alphabetical order. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 7 Order of Operation Calculators generally follow the traditional algebraic order of operations. Note that you can control the order of operation with parentheses. This family of calculators allows implied multiplication (no multiplication symbol is needed between the two objects) in many situations where there is no other interpretation. Just be careful with implied multiplication, because if there is any other interpretation possible, something else will happen. Finally the TI-85/86 family allows you to leave off final parentheses. It just assumes all “missing” right parentheses are needed at the end of the expression. It is very important to recognize the difference between the black subtraction key S 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 context determine the meaning. Notice on the screen that the negation is slightly higher and shorter. The subtraction operation takes two numbers as arguments, one before the key is pressed and one after. The negation operation takes only one number as an argument coming after the key is pressed. If you start a new command line by pressing the subtraction key ¹, the calculator assumes you wish to do a continuation calculation. Thus it assumes that you want to subtract something (yet to be typed) from the previous answer. You can also get the previous answer anywhere within the command line with , which is found above the negation key. There are many situations where you want to execute essentially the same command repeatedly. There are some nice editing features that make this easy to do. The command ,¡ found above the Í key causes the last command line to be recalled so that you can edit it. Pressing ,¡ several times allows you to go back to several previous command lines (limited by the size of some memory buffer). When you edit a previous command line, you do not need to move the cursor point to the 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, we have typed 11 Í and then pressed Ã 7 Í. As we repeatedly press Í we add 7 to the previous result. There is also a simple way to store the result of a computation for later use. The command is ¿ , and it will appear on the screen as an arrow →. You follow this command by typing a variable name. Notice that the calculator automatically switches TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 8 to ALPHA-entry so you do not need to press 0. Variable names must start with a letter and contain no more than eight characters. Variable names are case sensitive, so you may use upper and lower case. There is a way to “delete” a variable name, and you will want to do so to save memory when you no longer need the result. The command DelVar( followed by the variable name will do this, or you can delete individual items in the ˜ screen. It saves time if you store intermediate computations rather than copying down a number and retyping it later. Further, most people are lazy, and they copy down only a few of the decimal places. The “storing” operation saves the complete number with all significant decimal places for later use. Complex Arithmetic The TI-86 and TI-85 can handle complex arithmetic, and there is no turning complex numbers on and off (for they are always available). A complex number such as 2 + 5i is entered as an ordered pair (2,5). You can then add, subtract, multiply and divide complex numbers. The Š menu has other commands for complex numbers. The absolute value in the ‹ menu, NUM submenu has the traditional meaning for real numbers. 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 is always positive (unless the number is zero). Scientific Notation Even 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 out 5.7319 × 1025 which is difficult, they simply present 5.731925. You should use the same short-hand when you want to enter a number in scientific notation (avoiding multiplication by a power of 10). Use the Bkey where you want this symbol to be placed. Internally the calculator uses this notation, and 9.99999999999999 is the largest TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 9 number it can handle. If a computation results in a larger number, there will be an error message. 1⁻999 is the smallest positive number represented, and positive numbers smaller than that are rounded to zero. Exponents and Radicals Squaring is the key ¡. Further the ‚ keystroke raises a number to the negative one exponent. To raise a number to any exponent other than 2 or -1, use the › key. This command also works for negative and fractional exponents. Similarly there is a special command for square root ‡ (above H). For other radicals, use the MATH menu, MISC submenu command x√ such as the sixth-root of 64 below. Fractional Arithmetic The calculators in the TI-86/85 family are numerical calculators. They do not strictly do any symbolic operations such as fractional arithmetic. There is, however, a command that attempts to convert a numerical answer into some “nearest fraction” that can be useful if you want to compare your result to a simple fractional answer that might be given by someone working by hand. The command in the MATH menu, MISC submenu is Frac and it is very useful. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 10 Scatter Plots It is possible to plot an individual point in the coordinate plane using the command PtOn, either from the GRAPH menu, DRAW submenu or from the catalog. Issuing this command from the Graph screen, you get to select the point with the free-moving cursor (and Í). Issuing this command 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 the viewing window or regraph anything. A more permanent way to plot several points is to use a statistical plot. Suppose we 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 Doing this is significantly easier on the TI-86, so we will do that model first. To make sure that the statistical list editor is in the default configuration, press y v % then S, and select the command SetLEditor. Press Í to execute this command in the Home screen. We enter this data by pressing ,™ to bring up the STAT menu and by selecting & to begin to edit lists. Your lists displayed in this statistical list editor may or may not contain old data in the default lists named xStat, yStat, and fStat. The quickest way to clear out old data here is to do the following. Cursor up to highlight the name of the list, say xStat. Press Í to move the cursor down to the command line at the bottom of the screen. Press ‘ to empty out the command line, and then press Í to make this “empty” list the definition of xStat. Empty out yStat in the same way. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 11 Type the desired x-values in list xStat, and then type the corresponding y-values in list yStat. It is easy to delete mistaken entries and to insert additional entries with the { and o keys. After the data has been correctly typed, press ,k or - to get out of the list editor. Press ,™ to bring up the STAT menu and select ' to specify the statistical plot details. Select PLOT1. Highlight and then select items to match the following screen. The first Type is a scatterplot and the first Mark is a box. Notice that the menu line gives you appropriate selections when an item is highlighted above. We need to set the viewing window before we look at the graph of this statistical plot. Press 5' to bring up some quick ways to reset the window. For example, ZDATA will always resize the window so that you can see all of the data in a statistical plot. Here we have other reasons for preferring ZDECM so that pixel coordinates come in even tenths. After getting the graph, we check to see what viewing window settings were fixed by pressing WIND . TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 12 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 ( activates some kind of tracing action in the plot. For a statistical plot, we can see the coordinates of the points in the scatter plot as we cursor right and left. Trace Free-moving cursor estimate at x = 5.1 Suppose that the reader is asked to estimate the y-value when the x-value is 5.1. We can go back to the free-moving cursor point (5 and move a cursor key) to get a graphical estimate. Unfortunately the TI-85 does not have so many nice statistical features. Briefly, press the … key, and & to enter the statistical data. Then you must give the name of the two statistical lists. (For simplicity, accept xStat and yStat by pressing a twice.) While editing, the command CLRxy will clear out the current lists. Simply type the desired xypairs. Set the viewing window first by pressing 5'.( to get ZDEC. Then press … '& to get the SCAT DRAWing of this data. There is no choice of mark and no tracing for a statistical plot. You simply have the free-moving cursor by pressing one of the cursor keys. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 13 Function Graphing Make 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. Also let’s make sure that all of our function plots look the same by selecting the same formatting options. Press 5.' to get the graphical formatting screen, and change to matching the one on the right here. For example, let’s graph the function y = 3 x2 ! 12 x + 14 in the standard viewing window, as demonstrated on pages 54-55. Press 5% to get the Y= screen. Clear out any other functions that may be stored there, and make sure that no statistical plot is highlighted. Type the formula (using the key 1 or the menu key for x) in slot y1, press f ZOOM, and select ( ZSTD as shown here. Obviously this is not a particularly good choice for a viewing window for this function as noted in the text. One can now set a viewing window to see this parabola in a little more detail. The zoom command ZFIT (via ' . %) resets yMin and yMax so that the graph just fits with the screen for !10 # x # 10. To get rid of the menus at the bottom of the graphical screen press 9, and have them again press 5. Notice that we cannot see the x-axis any longer because the setting for Ymin is positive. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 14 Press the menu key for ZOOM and select ZPREV to get back to the graph before the last zoom operation. Then try ZOOM BOX to put a box around the parabola 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. 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 pixel coordinates just like 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 window while tracing. If you press Í while tracing, the window will shift so that the blinking pixel being traced moves to the center of the viewing window (called a Quick Zoom). If you trace all the way to the left or right edge of the graph and then continue to try to go farther, the window will shift to let you continue (called panning). Pressing 5. and selecting MATH FMIN from the menus allows the estimation of the minimum of the function on a subinterval. You input a lower bound and a upper bound on the subinterval and help the routine with a guess (which can be just moving the cursor point near where there is an apparent minimum). Considering y = 0.018 x4 ! 0.45 x3 + 2.93 x2 ! 1.5 x + 61.5 for 0 # x # 12, we get a plot similar the following. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 15 Tracing does not give integer x-values as we might want but the graph command EVAL allows us to evaluate the function exactly at specific x-values such as integers. You can also just type a number while tracing to evaluate the function at that number. Below we trace to an apparent maximum, use EVAL to find the largest value at an integer, and use the graphical MATH command FMAX to explore this function. By Trace By Value at Integer Maximum Solving Equations There are several ways to solve equations on this family of calculators. We begin with the techniques available in the graphical screen. Consider the task of solving for the x-intercepts and y-intercepts for the function y = 1000 x3 ! 15 x2 + 0.0002 from Example 1.5.8 (page 73). We type the formula in the graphical Y= screen and begin in the standard viewing window with ZOOM ZSTD as suggested in the text. Then we use ZOOM BOX several times to narrow in to a more appropriate viewing window as indicated below. ZSTD A more appropriate viewing window Using the Trace is merely a crude way of approximating the x-intercepts. To get more accuracy, one needs to repeatedly zoom in. Instead, use the graph MATH command ROOT to begin a numerical routine to solve for the zero or root of this function. The routine asks the user to give a left bound and a right bound for the subinterval where you desire to know the root. It is easy to give these bounds by moving the cursor point a little to the left and right of the apparent zero on the graph and then pressing Í. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 16 Then provide an initial guess for the zero, again by moving the cursor point to very near the apparent zero on the graph. The numerical routine works more rapidly if a more accurate initial guess is given. Repeat this procedure, giving different subintervals, to find the remaining roots. Finally the graph command EVAL followed by 0 a displays the yintercept. Remember that this procedure will only locate an intercept contained within your viewing window. The user might need to look at other larger viewing windows to be confident that this function has no other intercepts outside the ones we have considered. Panning and quick zooms might also help. There is also a graph MATH command ISECT to numerically find an intersection point 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 viewing window with !2 # x # 8 and !60 # y # 30 . This command prompts for the user to confirm which two curves to consider and to specify an initial guess to start its numerical routine. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 1 - Foundations and Fundamentals 17 Finally in the ,s there are ways to solve equations in interactive screen. For example, the x-value of the intersection point above is simply a solution to the equation 0 = 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 for the search with the bound. You begin the routine after things are set by selecting the menu item SOLVE. Graphing a Circle When 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 ydirection. The ZOOM command ZSQR will always change the viewing window to one with this property, 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 y = 64 − x 2 and y = − 64 − x 2 first in the standard window and then after ZOOM ZSQR. There is also a Circle( command to draw a circle in the graph DRAW menu and in the catalog, but drawn objects like this cannot be traced. ZSTD Then ZSQR Circle(0,0.8) Rational Function and Vertical Asymptotes Thus far, we have been using the graph format setting DrawLine to get nice TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs 18 connected graphs of the smooth functions considered. The calculator does this by plotting points (which are the ones you see when you trace), and then by turning on other pixels in the plot to make it look like those points are connected by short line segments. Most calculator and computer plots work this way by default. For rational functions, this connecting of the dots leads to a deceptive picture. It is better to convert to the DrawDot 18 2 format (or to at least look at both). Consider y = + − 5 in the standard x+2 x−3 viewing window. Notice that the near vertical lines at x = !2 and x = 3 appearing in the connected mode (where this function has vertical asymptotes) do not appear in the dot mode. DrawLine Format DrawDot Format Chapter 2 - Functions and Their Graphs Evaluating Functions After a function formula has been stored in the s Y= 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 P ( v ) = 0 .0178678 v 3 + 2.01168 v from Example 2.1.14 (page 130). We store this in y1 using the graphing variable x, and then type y1(20) on a TI-86. Notice that function notation will take precedence over implied multiplication. Instead, we must type 20 ¿x:y1 or … MISC evalF(y1,x,20) on a TI-85 (and this still works on a TI-86, too). Note the TI-85/85 is case sensitive, so make sure you get a lower-case letter y (via , m). TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs 19 The SOLVER allows us to quickly answer the question as to what velocity gives a power output of 500,000,000 watts. We can also get these results while looking at the graphical screen using the trace and value commands. Note that the you can actually type 20 while you are tracing to get the exact evaluation at x = 20. If we also plot Y2 = 500000000, then we can seek the intersection between 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 on a TI-86. In the Table Setup 6 [TBLST} we can choose between having the table entries automatically generated using the TblStart and !Tbl values or having the table entries determined by asking us. Now is a good time to mention the best way to choose a viewing window for a plot of a new function. First put the formula for the function in the s Y= editor. Then press Table Setup 6 [TBLST}and set the TblStart and !Tbl values so that we will get a table of function values where we think we want the interval [xMin, xMax]. The third step is to press [TABLE] to look at the function values. As we scroll through these function evaluations, take note of how we will need to set [yMin, yMax] if we stay with the original idea about [xMin, xMax]. Often we will decide to change even the xinterval as well after looking at a table of the function values. The fourth step is to set s [WIND] based upon what we observed in the table. Finally press [GRAPH] to see TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs 20 a plot that at least includes the pairs included in part of our table. Increasing and Decreasing, Turning Points We 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 graph MATH submenu commands FMIN and FMAX while viewing the graph. Consider f ( x) = 1 8 x − x + 2 from Example 2.2.8 (page 149). The graphs below are in the 3 2 standard viewing window, drawline format. Combinations and Composition of Functions Once 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 further function slots in the Y= editor. (Note TI-85 does not allow compositions or evaluation like this.) We can only plot or evaluate these. There is no symbolical operation to simplify the new functions created by these operations. Consider f(x) = 2 x2 + 4 x + 5 and g(x) = 2 x + 1 from Example 2.3.5 (page163). Inverse Functions The commands 5 {DRAW} DrawF and Drinv plot a non-interactive graph of a function and the inverse of a function. Notice that this command for the inverse is really just interchanging the x-coordinates and y-coordinates for plotting purposes. The function does not need to be one-to-one and may not have a true functional inverse. Still the plot is right when the function has an inverse. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs DRAW Commands Standard Window 21 Square Window Graphing a Family of Functions A 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 see the functions in the family f(x) = a x2 which are plotted in Figure 2.71 (page 183) by using the list {-2, 0.5, 1, 4} in place of the parameter a. Note that you can find the braces {} in the “ menu. We use ZOOM ZDEC to get a nice window. Piecewise-defined Functions Piecewise-defined functions can usually be handled on a TI-86/85 using logical tests. The — menu provides the various inequality and equality symbols. A logical test on a TI-86/85 evaluates to 1 if it is true and 0 if it is false. We use this to “zero out” parts of a formula when we do not want that part to contribute. Consider R− x + 6 x f ( x) = S T x − 3, 3 2 − 9 x + 4, x<3 x≥3 from Example 2.5.5 (page 187). This is typed as y1=(−x^3+6x-9x+4)(x<3)+(x-3)(x≤3) Notice that in the DrawLine format, the nearly vertical line between dots connects the two pieces where it is not appropriate. The DrawDot format does not do this (although it also leaves dots within pieces unconnected as well). You can also get Dot as a STYLE in the TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs 22 Y= Edit screen. The difficulty with this way of representing piecewise-defined functions is that all of the pieces must be defined for all numbers x in the x- interval to be considered, even when you 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) outside of this subinterval because the middle piece is not defined there. A “fix” for this particular example is the following. Y2 = (x+8)(x<−4)+(√(abs(16-x2)))(X≥−4 and x≤4)+(x-8)(x>4) Least-Squares Best Fit The TI-86/85 provides several different regression fits for numerical data, including using linear, quadratic, cubic, and quartic polynomials. We demonstrate here only a linear fit on a TI-86. See the TI-85 manual for the slightly different steps (particularly for editing statistical data). Consider Table 2.10 giving U.S. health-care expenditures (in billions 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 hand computations). Here we show that there is no need on the calculator to do this. Thus our regression function will be different (having a different definition of the variables). Our TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 2 - Functions and Their Graphs 23 graph will have the actual years as the first coordinate, and to evaluate the regression function for 2003, we will simply need to enter in the variable 2003 (not t = 18). As we did in Chapter 1, enter the years in list xStat and the expenditures in list yStat in the ™ EDIT list editor. As we did before in Chapter 1, turn on a statistical scatter plot of this data and use ZOOM ZDATA to size the viewing window in an appropriate manner for this data. Press ™ CALC LinR to have this regression performed. If you give no arguments, the command LinReg assumes data will be in lists xStat and yStat. The more complete way to issue the command, however, is LinReg xlist,ylist,y# where the resulting equation will go immediately to the function slot y# . After this computation, the coefficients a and b and the formula for the regression equation can be found in v STAT to be used later. Notice that our result is TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 3 - Polynomial and Rational Functions 24 y1(x) = 59.118 x + -116947.69 which does not agree with the E(t) = 59.118t + 401.54 given in the text. When we use the formula 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 Functions A graphing calculator is very nice for investigating polynomials of degree three or higher. We use the same techniques for setting viewing windows, finding zeros, and finding turning points as before. The added feature about polynomials is that we have a few theorems to help us know when we have found enough zeros or turning points. Here is one trick for making the evaluation of a high degree polynomial more accurate and the graphing more rapid. We can algebraically rewrite a polynomial in several equivalent 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 our calculator rather than the first, we can avoid using the “power key” › which is quite slow for repeated computations and otherwise reduce the number of arithmetic operations required in evaluation. On virtually all graphing calculators, entering this polynomial in the second way will cause it to plot in about half the time as entering it the first way. Polynomials can get very big, making the ZOOM ZFIT a very attractive option after you have set the x-interval for the desired window. We use this to plot the above fifth degree polynomial. (Usually you will want to go back to the window screen and readjust the Yscl as was done below after the window is set by ZFIT.) TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 3 - Polynomial and Rational Functions 25 Then to study zeros and turning points you will probably want to zoom out a little to check end behavior and to zoom in to better see zeros and turning points. A TI-86/85 graphing calculator has no special features for symbolic operations with polynomial multiplication, division or complex roots. There is procedure on these calculators called [POLY] that can find the roots of polynomials (real and complex). Here we demonstrate with two different polynomials. First we consider Example 3.3.1 (page 245) f ( x ) = 5 x 4 + 8 x 3 − 29 x 2 − 20 x + 12 . Next we consider Example 3.3.6 (page 251) g ( x) = x5 − 2 x 4 − x3 + 4 x 2 − 2 x − 4 . TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 4 - Exponential and Logarithmic Functions 26 Note that any numerical root finding algorithm will have trouble with a double root. Here the polynomial g(x) has !1 as a double root. This is approximated by the two complex roots −1 ± 3.05044259084E-7i , each with very small imaginary part. This is not a 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 Functions For a detailed look at vertical and horizontal asymptotes for rational functions, it is nice to zoom in and out in one direction at a time. Also don’t forget that the dot mode x − 2x + 2 2 generally is best for this family of functions. Consider f ( x ) = 2x − 4x 2 from Example 3.5.4 (page 274) on various windows . !3 # x #5, !5 # y #5 Overall view !25 # x #25, 0.4 # y #0.6 1.9 # x #2.1, !50 # y #55 Highlighting end behavior Vertical view near x = 2 Chapter 4 - Exponential and Logarithmic Functions Exponential Functions We 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). Note that you get the braces {} surrounding the list from the “ menu. Try the trace on this plot. y1={4-1,3-1,2-1,2,3,4}^x TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 4 - Exponential and Logarithmic Functions 27 Two special exponential functions are provided on the keyboard, 10x and ex, and you should use these rather than the power key for more accuracy. You can use the natural exponential keystroke to get the value of this number e with e^(1) 2.718281828. Using these methods will be better than typing these digits because even the guard digits you cannot see will be correct. Logarithmic Functions The two special logarithmic functions provided on the keyboard give common logarithms, «, and natural logarithms, µ. Use the Change-of Base Formula (p. 326) to work with logarithms in another base in terms of one of these special ones. log u ln u log a u = = , a ≠ 1, u > 0 log a ln a Regressions Involving Exponentials and Logarithms The ™ CALC menu offers a number of regression options that involve families of exponential and logarithmic functions. The preliminary steps for these regressions are the same as for linear regression above in Chapter 3. You simply select and plot a different regression fit. You can even plot several on the same screen and decide visually which seems to be the best fit. LnR ExpR PwrR LgstR a + b ln(x) a bx a xb c 1+ a e −b x (only on TI-86) For example, consider the data from Table 4.7 (page 344) describing a state deer population (in thousands) since 1999 (t = 0). We show how to obtain an exponential fit for the data. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 4 - Exponential and Logarithmic Functions Year (since 1999) Population (in thousands) 28 0 1 2 3 4 5 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 exponential regression, and save the regression equation in a function slot. Finally, compare the scatter 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 can also follow the instructions in the “Calculator Keys” box on page 346 to convert the base of the exponential function given by the calculator to the natural base e. Remember that immediately 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. ( P (t ) = 9992.407418 (1.149206874 ) = 9992.407418 e t = 9992.407418 ( e 0.1390720296 ) = 9992.407418 ( e ) t ln (1.149206874 ) ) t 0.1390720296 t The logistic regression is a very difficult computation. The routine in the TI-86 may fail to converge. It seems to have problems with large data. In particular, it did not seem to work for this data set. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 5 - Trigonometric Functions 29 Chapter 5 - Trigonometric Functions Angle Measurement The TI-86/85 has an angle mode setting of either Radian or Degree in the mode screen. We will experiment here with both settings. Pressing ‹ ANGLE brings up a menu with further angle commands. The first ° causes the number before this symbol to be interpreted as degrees, regardless of the angle mode. The second ' is the separator for the degrees-minutes-seconds notation and the third gives radians, again regardless of the angle mode. Note that there is also a special keystroke for Ä above the › key. Assuming degree mode setting, expressions given in degrees-minutes-seconds (DMS notation) will be converted to decimal degrees. The command ‹ ANGLE DMS converts something in decimal degrees into DMS. Expressions designated in radians with will be converted to degrees. In degree mode, the degree symbol ° alone does nothing. Assuming radian mode setting, expressions given in degrees-minutes-seconds (DMS notation) will still be converted to decimal degrees (but with no indication to interpret answer in degrees). The command ‹ ANGLE DMS still converts something in decimal into DMS, interpreting the decimal as decimal dgrees. Expressions designated with only the degree symbol ° will be converted into radians. In radian mode, the radian symbol does nothing. Sine, Cosine, and Tangent Function Keys The 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 to override the angle mode should mean that a student does not need to change a mode setting to switch back and forth between degrees and radians for simple trigonometric TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 5 - Trigonometric Functions 30 computations. The most common error students make working with these functions is to be in the wrong angle mode. A goal should be to know enough about trigonometric functions so that you can immediately recognize when you start to get answers appropriate for the wrong angle mode. On the TI-86/85, no parenthesis is automatically provided with trigonometric functions, and the order of operation may give you surprising results (i.e. trigonometric operations take precedent over multiplication and division). Degree Mode Radian Mode The inverse trigonometric functions [SIN-1] [COS-1] [TAN-1] also depend upon the angle 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 in degrees, you must be in degree angle mode. Degree Mode Radian Mode Plotting the Sine, Cosine, and Tangent Functions Since graphing calculators are used to plot trigonometric functions so often, a special viewing window is provided that is frequently appropriate for these functions. The command 5 ZOOM ZTRIG resets the viewing window to Degree Mode Radian Mode !472.5 # x # 472.5, Xscl = 90, !4 # y # 4, Yscl = 1. −8.24668071567 ≤ x ≤ 8.24668071567, Xscl = π 2, RS T − 4 ≤ y ≤ 4, Yscl = 1. The unusual endpoints for the x-interval give nice fractions of 90° or B radians as pixel coordinates for tracing. Below are examples in radian angle mode. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 5 - Trigonometric Functions Sine DrawLine Cosine, Tangent 31 DrawDot Cosine, Tangent Families of Trigonometric Functions We can plot several functions in a family again by using a list for one of the parameters. First we store 1 in variables A, B, and C. Then we create a list L1 ={0.5, 1, 2, 4}. 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=L1*sin(B*x+C) y2=A*sin(L1*x+C) y3=A*sin(B*x+L1) Cosecant, Secant, and Cotangent Functions There is no keystroke for the remaining trigonometric functions on the TI-85/86. You need to know the fundamental identities for how csc x, sec x, and cot x are related to sin x, cos x, and tan x (namely that they are reciprocals). For csc x type (sin x)ñ or 1/sin(x). For sec x type (cos x)ñ or 1/cos(x). For cot x type (tan x)ñ or 1/tan(x). We will leave as a challenging exercise the task of determining what to do for the inverses of the cosecant, secant, and cotangent functions. You are well advised, however, to avoid the 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 Tangent Again, 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 5 ZOOM ZTRIG still gives a reasonable viewing window, although we may only be using a small TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 6 - Trigonometric Identities and Equations 32 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=sin x y2=cos x y3=tan x Chapter 6 - Trigonometric Identities and Equations Graphical Check of Equations When 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 a contradiction. Generally we graph the two sides of the equation separately and look for intersections. When you trace, use the up and down cursor keys to toggle between the two different sides. Example 6.1.1 (p. 462) Example 6.1.2 (p. 463) 2 sin x = 2 - 2 cos x (sin x + cos x)2 = 1 + sin 2x For potential identities, it is actually more convincing to look at tables of the two expressions evaluated for the same x-values. The graphs may look the same but merely be close. The table entries appear to be exactly the same. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 6 - Trigonometric Identities and Equations 33 Example 6.1.2 (page 463) continued Example 6.1.3 (p. 463) 2 - sin x = cos x Conditional Trigonometric Equations We 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 first plotted both sides, then we can compute intersections of the two separate curves in the graph. Just make sure that your guess is very close to the intersection you want. If we rewrite the equation so that one side is zero, we can seek a zero on the graph instead as in y3 below. y1 = cos 2x, y2 = 2 cos x, y3 = 2 cos2 x - 2 cos x - 1 Finally, if we manage to reduce the problem to something such as cos x = 1− 3 , 2 then we can use [COS-1] and our knowledge about the reference angles to solve for x in the interval [0. 2B). TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 - Applications of Trigonometry 34 Chapter 7 - Applications of Trigonometry Complex Numbers Revisited The rectangular form for representing a complex number a + b i, is to put the real and imaginary parts as an ordered pair separated by a comma (A,B) on a TI-86/85. The trigonometric form for representing a complex number is r (cos 2 + i sin 2) (see page 538). This is available too on a TI-86/85 in a slightly different notation called the polar form r ei2 . The variables r and 2 have exactly the same meaning in the trigonometric 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 on page 555 for more detail about what is called Euler’s formula.) The polar form is represented again by an ordered pair, but now separated by an angle symbol (R∠ θ). Note that the angle symbol is a , function above the comma O. Note that the angle mode (radians or degrees) affects how the angle 2 will be given in the polar form. Here we assume radian mode. The command Š åRect converts to the rectangular complex form. The command Š åPolar converts to the polar complex form. Note that you can type in complex numbers in any form. Often the resulting complex number is too long to see all of it on the screen at once. Just press the right and left cursors to see the result before beginning to type the next command line. The modulus is obtained by the Š command abs. The square root command and the power command (to get other nth roots) give principal roots (not all nth roots). For example, the fifth roots of 3 e1.2 i can be found from the principal fifth root x = 1.24573094 e0.24 i = r ei 2 given by the calculator by repeatedly adding 2B /5 to the argument 2. Thus we get the collection of fifth roots to be { iθ re , re c i θ +2π 5 h , re c i θ +4π 5 h , re c i θ + 6π 5 h , re c i θ + 8π 5 h } . Note that the calculator program in Exercise 7.4.50 (page 554) will run almost exactly as written there on a TI-86. The only slight changes are that the command to clear the TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 - Applications of Trigonometry 35 drawing is ClDrw and labels must be given a letter (not a number). While the oldfashioned command IS>(K,N) indicating to “Increase K by 1 but Skip the next command if K > N” is still there on these newer calculators, there are now better ways to loop. Executing NGON with n = 8 in window −1.6997 ≤ x ≤ 1.6997, − 1 ≤ y ≤ 1 Polar Coordinates On a TI-86/85, the conversions between rectangular complex form and polar complex form are exactly the conversions from rectangular coordinates and polar coordinates. The calculator has chosen to convert the rectangular representation (0, 0) to the polar (0; 0). It gets a unique polar representation for rectangular coordinates other than the origin by selecting r > 0, 0 # 2 < 2 B. Press 5 FORMAT, and you will find the first formatting option is to select rectangular graphing coordinates RectGC or polar graphing coordinates PolarGC. This format option will determine the coordinates that appear at the bottom of the graphical screen in all graphing modes. Plotting Polar Equations On the l screen, move from function graphing to polar graphing by selecting Pol. On the 5 FORMAT screen, select polar graphing coordinates PolarGC. Then press 5 r(θ)= to see 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 obtained TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 - Applications of Trigonometry 36 either by pressing the menu key % while in the r(θ)= editor or by getting it as a Greek character from the menu at any time. In addition to setting the x-range and y-range on the 5 WIND screen, you now must also set 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 ZDECM to get a decimal, equally scaled viewing window and also to get [0, 2B] for 2, with 2step = B/24 so that we hit favorite multiples of B as we trace. The graphical MATH menu no longer contains an “intersect” command, and there is a good reason for this. An apparent point of intersection of two polar equations can occur because of one representation of that point in one equation and a different representation of that same point 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 coordinates giving the point on each equation and by turning on rectangular graphing coordinates as well, we can roughly compute the following table describing these three points and how they solve each equation. (x, y) (1.6, 0.8) (0.3, 0.7) (0, 0) r = 2 cos 2 (1.8, 0.4) or (-1.8, 3.5) (0.8, 1.2) or (-0.8, 4.3) (0, 1.57) or (0, 4.7) r = 1 + 2 sin 2 (1.8, 0.4) (-0.8, 4.3) (0, 3.7) or (0, 5.8) We can get more accuracy on a TI-86/85 with the interactive s, using this initial graphical work for starting guesses and for setting the equations to be solved. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 - Applications of Trigonometry 37 Vectors There is a special vector data type and a ‰ menu with special vector operations on a TI-86/85. You can use the ‰ editor or you can type vectors in the Home screen between square brackets separating the elements of the vector by commas. Vector addition, vector subtraction, and the multiplication of a vector by a scalar can be computed. Included in the mathematical functions and operations for vectors are commands for finding a unit vector in the same direction as a given vector, for finding the norm of a vector, and for converting between vectors and lists. It is possible to use the drawing command for a line segment to get a rough sketch of the magnitude of a vector and to picture the idea of one vector added to the end of another. One way is to get the command Line( from the v CATALOG for use in the home screen. Line( expects as argument the coordinates of the starting point and ending point. Unfortunately there is no simple way to put an arrow at the end of any of the line TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 8 - Relations and Conic Sections 38 segments to indicate direction. Here we draw line segments to represent P(-1, -2), Q(-3, 1), and P Q with the initial point of each vector at the origin. The other way to draw lines is to use the LINE command from the DRAW menu while in the graphical screen. We add another line segment for P Q putting the initial point at the terminal point for P(-1, -2) by using the cursor point to place the beginning and ending point for the line segment.. Our viewing window is from ZDecimal and we have RectGC as a format so that then we see rectangular coordinates. Chapter 8 - Relations and Conic Sections Graphing Relations in Pieces There 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 4 x + 9 y = 36 as y = ± 2 2 b36 − 4 x g / 9 2 . Just to highlight some potential difficulties, we select the window with ZOOM ZTRIG and plot both the upper (+) and lower (-) parts of the ellipse as separate functions (in function graph mode, RectGC). TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 8 - Relations and Conic Sections 39 Notice that the upper and lower parts of the ellipse do not quite meet. Each of these function 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 = 2.8797932658. The next pixel to the right has coordinate x = 3.0106929597, and in this column of pixels there is no plot. We do not land exactly on x = 3 as a pixel coordinate, where both Y1 and Y2 would evaluate to zero. Using ZOOM ZDECM here does give such a pixel coordinate at all of the integers as well as other decimal values. Plotting Parabolas A parabolas 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 or left instead, we can either plot two separate pieces (where we can trace on each piece) or we can switch the variables x and y and use the DrInv command. From Example 8.2.7 (page 619) consider x + 1 = − 1 2 ( y − 2 ) 2 . These plots are in a standard viewing window. Tracing Function Free-moving Cursor Near Drawn Object Plotting Hyperbolas In 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 pieces possibly not meeting because of pixel coordinates not exactly hitting the vertices. Consider Example 8.4.3 (page 646). TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 8 - Relations and Conic Sections y1 and y2 40 y3 and y4 Plotting Parametric Equations A TI-86/85 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 parametric graphing mode, the 5 E(t)= edit screen provides the graphing variable t as a menu choice. For the viewing window below, we started first with the standard viewing window ZOOM ZSTD, and then did a ZSQR 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 the right 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 equations as well. We set the t-interval for the parameter as well as the bounds for the axes. The setting Tstep determines the plotted points (which then can be traced). In the DrawLine graphing mode, small line segments are drawn between the plotted (traceable) points. If Tstep 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. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 - Systems of Equations and Inequalities 41 Chapter 9 - Systems of Equations and Inequalities Matrices A TI-86/85 can store a matrix to a variable name. While you can enter very small matrices in the home screen, it is more convenient to use the matrix editor. The , function keystroke ˆ brings up the matrix menu. The editor on a TI-85 is not as visual as the editor on a TI-86, but both are functional. The square brackets can be used in the home screen to create small matrices. An error message will appear if you try to add, subtract, or multiply matrices which do not have 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, this command can be used to form the augmented matrix using the coefficient matrix and the right-hand side of the equation. Gaussian Elimination All 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. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 - Systems of Equations and Inequalities 42 If you want, you can overwrite the matrix you started with or you can store in another matrix. If we follow a matrix name by the row and column in parentheses, we can isolate an individual 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 matrices can 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 a matrix. We demonstrate this on matrix A below. The command is ref( to convert to a row-echelon form equivalent to the starting matrix. Gauss-Jordan elimination is done by the command rref( to convert to the unique reduced 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. Identity Matrices, the Inverse of a Matrix, Determinants You can quickly get an identity matrix (with ones on the diagonal and zeros elsewhere) with the matrix operation ident by simply giving the size desired for this new square matrix. For a square matrix which has an inverse, the key — gives the inverse. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 - Systems of Equations and Inequalities 43 There are three ways to solve a system of linear equations such as 2x + 3y + z = 6 4 x − y + z = −3 x + y + 1 2 z = 1 involved in Figure 9.6 (page 749). One way is to form the augmented matrix [A|B] and apply rref to it. The second way is to find the inverse A-1 of the coefficient matrix A and multiply it times the right-hand side B. We can also find the determinant, say of matrix A. The third way is to use the simultaneous linear equations solver t. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 - Systems of Equations and Inequalities 44 Systems of Inequalities Consider Example 9.7.4 (page 772) asking for a graph of this system of inequalities. x + 2y ≥ 2 − 3 x + 4 y ≤ 12 Enter each inequality as a function equality solved for y, and select the style (shade above or shade below) to match Figure 9.30 using the window !9 # x #9, !6 # y #6. These function graphing styles are only available on a TI-86. On a TI-85, the best you can do is to plot 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 as in the text. A suggestion is to reverse the shading which amounts to shading the part of the 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 “white area” to get a picture similar to Figure 9.31. Using the graphing style to “shade above” or “shade below” will only work for inequalities that can be solved for y. This is best, if we can do it, because we can trace on the bounding curves for the region and find intersections to active function graphs. In other situations we use DRAW Shade to shade between a lower function and an upper function over a perhaps more limited x-interval. (This Shade command is found on the TI-85.) 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. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 - Systems of Equations and Inequalities 45 Example 9.7.5 (p. 772) x+ y≤4 −2 x + y ≤ 1 y ≥ −1 x≤2 Shade to “Cross Out” Example 9.7.7 (p. 776-7) x2 y2 − >1 9 4 Shade Desired Region Linear Programming A TI-86/85 can be great aid in identifying the feasible region, locating vertices, and evaluating the objective function at the vertices. We demonstrate this here by working Example 9.8.1 (page 789-90) M inim ize K = 100 x + 60 y subject to 250 x + 250 y ≥ 750 0 . 6 x + 0 .06 y ≥ 0 .72 12 x + 60 y ≥ 60 x ≥ 0, y ≥ 0 Note that we can handle the last two inequalities (x $ 0, y $ 0) by simply setting the viewing 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 intersection TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 10 - Integer Functions and Probability 46 point using the intersection command, and we return to the home screen to evaluate the objective function using the coordinates of the intersection point. Chapter 10 - Integer Functions and Probability Sequences The TI-86/85 family of calculators has no sequence graphing mode. Thus it will be a little more difficult to do the same activities that we do with the other families of calculators. We can create a list of a finite number of terms in a sequence given by a formula using the “ OPS seq command. You use any variable name as the index for the sequence in the formula, give the variable name, the start, and the end (with the stepsize as an optional last argument, default stepsize being 1). Suppose we wish to see a plot of the sequence an = 4 + 6(n !1), n = 1, 2, ÿ which is similar to a TI-83 plot of this sequence in its sequence graphing mode. We start on a TI85/86 by creating two lists. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 10 - Integer Functions and Probability 47 seq(N,N,1,35) ÷ NN and AA = seq(4+6*(N-1),N,1,35) Then we create a statistical scatter plot with these two lists. ÷ AA ZOOM ZDATA If a sequence is defined recursively, then we need to write a short program to create the second list representing the terms in the sequence. Below is a sample for Example 10.1.11 (page 815) an = 2an-1 + 5, a1 = 3 done on a TI-86. Begin editing 7 EDIT Short program Prompt within program Resulting sequence list Statistical Plot Setup ZOOM ZDATA TI-86/TI-85, Precalculus 7 NAMES to run Integer list Sequence Plot © 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 10 - Integer Functions and Probability 48 Series The simplest way to compute a series is to use the “ OPS commands seq and sum together. Consider parts a. and b. of Example 10.2.2 (page 822). 5 ∑2 k =1 6 k ∑i 2 i =1 We can also write short programs with a For-End loop to do these computations. Permutations, Combinations, Random Numbers Many questions in probability involve the use of factorials, permutations, combinations, and experiments with random numbers generated by computer or calculator. Commands for these operations can be found in the ‹ PROB menu. Use the built-in commands for nPr and nCr rather than the formulas involving factorials because it allows n to be larger. For n $ 450 the factorial computation will overflow on a TI-86/85 but you can still compute further permutations and combinations. TI-86/TI-85, Precalculus © 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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