Chapter 5
Contents
1. 3 4 6 ir 132 CHAPTER 5 Expanding your graphing skills Pieti Filekte Flats FARAEOLA Line Figure 5 21 Graphing a parabola and annotating it PEAK and PARABOLA are drawn with the Text tool the circle is a Circle and the arrow is three Lines The left side shows the equation and the middle shows the graph before being annotated always turn the axes on and off and modify other graph format options using the keys Now let s try some annotations a Try circling the peak From the graph press to open the Draw menu and scroll down to 9 Circle Move the cursor over the peak press to set the circle s center and then move the cursor to a point where you want the edge of the circle to be and press again A circle 2 Write out PEAK as in figure 5 21 Go to 0 Text in the Draw menu and then use the arrow keys to move the cursor Place it at the top left corner of where you want the word PEAK to appear and type PEAK Remember that to type a string of letters you press to engage Alpha Lock and then press the keys that have the letters you want printed over them 3 You can type PARABOLA without returning to the Draw menu Move the cursor to a new position with the arrow keys and type PARABOLA 4 Create the arrow pointing to the parabola in figure 5 21 Go to the Draw menu and select the 2 Line tool For each line move the cursor to the first endpoint press ENTER move the cursor to the second endpoint and2. and choose 9 GDB Figure 5 27 shows an example of storing a GDB messing with Graph Format set tings and graphed equations and then recalling the original GDB By recalling GDB4 the initial set of functions and settings from the left in figure 5 22 is restored undoing the changes I made to the graph format and the window settings And now we ve reached the end of what you need to know for advanced usage of your calculator s drawing and graphing features From here on we ll focus on advanced 140 5 6 CHAPTER 5 Expanding your graphing skills Loam wad T i iadd grid O Storesoe 4 Gerecslleoe 4 4 Figure 5 27 Starting with the graph on the left the StoreGDB 4 command stores the graphed equations and graph settings to graph database GDB4 After I turn the axes off and the grid on and change the window settings the graph changes to what you see in the center The Recal1GDB 4 command restores it to the right view just as it was when the StoreGDB 4 command was issued math features although some of the discussion such as statistics will include use of plots and graphs on the graphscreen Summary In this chapter you learned advanced skills for graphing and drawing We started with the three graphing modes that chapter 3 didn t explore Parametric mode Polar mode and Sequence mode You learned how Parametric mode lets you express and graph functions that would be impossible in Function or Rectangular mode You expl
3. Graphs a function just as if you had entered it in the Y menu This function works only with rectangular function Y equations even if the graphing mode is set to one of the other modes Like DrawInv it takes a single argument an equation to graph For example DrawF V X would be equivalent to setting Yi V X Shade Draws a solid or hashed shade between two functions or y coordinates It takes at least two arguments the two functions or y coordinates bounding the shaded area For example Shade X 10 shades the area between y x and y 10 in solid black You can also choose minimum and maximum x coordinates on the shaded area so Shade X 10 0 2 only shades between x 0 and x 2 Except for DrawInv s use as a way to graph X equations you probably won t find your self using these three tools that often Nevertheless for the sake of completeness I d like to show you a brief exercise that uses all three of these drawing tools You ll use DrawF and DrawInv to graph Y 3sin X and X 3sin Y and then shade between Y 3sin X and Y 3sin X 3 using the Shade command To set up press and change your calculator to Function mode if it s not already in it Turn on the axes from the Graph Format menu 2nd_ zoom if they re not already on and select 6 ZStandard from the Zoom menu Finally if you still have anything on the graphscreen clear any entered equations from the Y menu 5 5 Saving graph settings
4. a new value By default Tstep is 7 24 You can adjust the Tmin Tstep and Tmax values to change how a parametric graph looks PI show you how changing Tmax can turn the circle you graphed into a semicircle GRAPHING A PARTIAL CIRCLE You may recall from trigonometry or geometry that a circle is 2m radians or 360 around Because by default Tmin 0 and Tmax 27 your calculator plugs all the values necessary into your Xir 6cos T and Yir 6sin T equations to draw a full circle of radius 6 If a full circle is 27 it stands to reason that half of a circle is just m Press noo move the cursor down to Tmax and change the value to n which you can type with a The left screenshot in figure 5 5 shows what you ll see Press to set the value and the screen will transform into what you see in the center in figure 5 5 Press to graph the result a semicircle that should match the right side in fig ure 5 5 Your calculator plugs in T values from T 0 to T 7 and you get the first half of the circle You can change Tmin to 7 and Tmax to 27 if you want the other half of the circle You can also try making Tstep smaller and larger Notice that if you make it larger the circle gets rougher because your calculator is plugging in fewer T values Make it smaller and the circle gets smoother but also takes longer to graph If you set it low enough you might even be able to draw polygons We can revisit parabolic or projectile motion from chapter 3
5. as explained in the Polar graphing in a nutshell sidebar the Y and Window menus will change to reflect the new variables and equations To get you accustomed to polar graphing let s start with an example of the easiest graph you can draw in Polar mode a circle Ironically it s one of the hardest things to do in Rectangular mode requiring two equations and careful tweaking of the AX vari able Polar mode plugs in 9 values from 0 to 27 to any equation you enter gets a radius and plots a point a distance rfrom the origin in the 9 direction If you enter a constant number like 4 2 or 8 for a polar equation then your calculator will draw a circle of that radius Let s try ri 7 The first thing you need to do is make sure your calculator is in Polar mode Go to the Mode menu move the cursor to POL and press ENTER You should see what the left CHAPTER 5 Expanding your graphing skills NORMAL SCI Emni Floti Filekte Floks FLOAT Ge Se i Br Figure 5 11 Setting Polar mode and graphing your first polar equation The left shows switching to POL in the Mode menu The center screenshot displays a simple equation to graph and the right is the resulting circle Polar graphing in a nutshell To switch to Polar mode press MODE move the cursor to POL and press ENTER When you go to the Y menu you see six single functions you can enter from Yi tO re m When you press the key in POL it types a 9 instead of an x because
6. default O goes from 0 to T so you get only one revolution of the spiral around the ori gin To create more of the spiral you need to make Omax larger First press and change Omax to 67 This will give you three revolutions of the spiral which should look much more interesting To make the spiral look even better switch to a square window press and select the 5 ZSquare option if you didn t already have a square window Now tap once more and you should see the attractive graph in figure 5 13 You can reverse the direction of the spiral by setting r 0 r instead You can make the spiral tighter or looser by adjusting the scaling factor here 1 7 You can modify Omin Ostep and Omax to see more or less spiral or to make your calculator graph a smoother spiral I encourage you to experiment and see what you can do with the graph Floti Floke Floats Figure 5 12 An initial attempt at drawing a spiral With the default graph settings you get only one section of the Spiral from 9 O to 0 27 124 5 2 2 CHAPTER 5 Expanding your graphing skills Let s examine another polar exercise a family of graphs that creates shapes called polar roses Polar example polar rose The polar rose is a family of graphs that as you might guess from the name are drawn in Polar mode and look like roses Specifically they re centered on the origin and have Figure 5 13 Your final spiral from 0 0 to 0 6r Notice eral e
7. has a lot of options nMin and nMax con trol the range of n values that are plugged into the sequences There s no such thing as nStep because n always increases by 1 There are also settings for the coordinates of the edges of the screen as in every graphing mode m The PlotStart and PlotStep options in the Window menu control which values of the sequences are shown on the graph This won t change the values that are calculated though which are controlled by nMin and nMax 126 5 3 1 CHAPTER 5 Expanding your graphing skills RECURSIVE A recursive sequence is one where to find the value of the sequence term Un you need to calculate other terms in the sequence as well Recursive means that to calculate those other terms you need to find still more terms and the chain continues The Fibonacci sequence we ll examine in section 5 3 2 is a popular series or sequence for teaching recursion Let s start with the easier of our two exercises the sequence of squares This will give you a good introduction on how to enter sequences and graph them as well as how you can use the handy Table tool to see successive values of sequences Sequence example a sequence of squares Sequences can range from the simple and easily understood to the very complex We ll start near the easy end with a sequence containing the squares of the positive integers Its equation is Un n SO m 1 w 4 us 9 u 16 and so on It s particu
8. larly simple as far as sequences go in that you can calculate any term without know ing any other term To calculate w you plug 9 into un n to get w 81 Start by switching your calculator into Sequence mode as described in the Sequence graphing in a nutshell sidebar Next head to the Y screen and enter the u n equation for this sequence u n n Remember the key types the independent sequence variable n for you As you can see in figure 5 15 on the left you should leave u mMin blank for now because it s only used for a recursive equation You might also remember from chapter 3 that the three dots next to u n mean that you re graphing this sequence as disconnected dots instead of a line When you press GRaPH you ll see the results in the center in figure 5 15 Of course if you don t have the ZStandard window set things might look different but choosing 6 ZStandard under the Zoom menu will swiftly fix that Because this window is only big enough to show three of the points in this series you might want to adjust the win dow Intuition would tell you that you need to see more of the graph above the cur rent window so you should increase Ymax The right graph in figure 5 15 exemplifies the window changed to Ymin 0 and Ymax 100 Floti Flokte Filoks a n Rte oC eee wtp wooMing n 1 u f 1 ier er weehlina Figure 5 15 Graphing a sequence of squares On the left entering the u n equati
9. lessons about using ZoomFit to see the graphed function better and the Table tool to get exact values GRAPHING THE FIBONACCI SERIES Just to enter the Fibonacci series in your calculator s Y menu you need to learn two new things First you need to learn how to refer to other terms in a series while you re Graphing sequences 129 writing a recursive function definition as u n Second you need to know how to tell your calculator the base cases for the recursive series The first problem is defining the function for the recursive series You already know that each term is u n and your calculator plugs in values for n What if you want to refer to the previous term which in math notation is uni Use u n 1 typed with As the Sequence graphing in a nutshell sidebar explains is the u equation is the n variable in Parametric mode and the rest is stuff you ve seen before If you want to refer to the term before that use u n 2 and so on For the Fibonacci example enter u n u n 1 u n 2 as shown on the left in figure 5 18 FORWARD REFERENCES On your calculator you can t refer to terms after the current term in a sequence such as um via u n 1 This is invalid syntax If you try you ll get an ERR DOMAIN error You can only refer to terms before the current term Next you need to enter the base cases Ifyou have one base case you can enter it as a single number next to u mMin and v n
10. press again The tool you select stays active until you select another tool so you don t need to return to the Draw menu before each new line The arrow in figure 5 21 is three lines While you were drawing you might have been frustrated that the coordinates at the bottom of the screen were getting in your way If you don t mind them while you re drawing but you want them to go away when you finish drawing press CLEAR Don t worry it won t clear the screen At worst you might accidentally quit to the home screen at which point you need only press to return to the drawing If you don t want coordinates to show at all while you draw select CoordOff from the Graph For mat menu zoom If you re on a TI 84 Plus C Silver Edition you could press in the middle of drawing any shape to access the Style menu as indicated by the onscreen Style button above the key CLEARING THE GRAPHSCREEN CLRDRAW The 1 ClrDraw command from the Draw tab of the Draw menu clears the entire graphscreen If you have the axes or grid enabled it redraws them as well and then rerenders any functions you ve graphed Press GraPH and then go to the Draw menu 5 4 2 Drawing on graphs 133 with Prem and choose 1 ClrDraw If you re on the homescreen you can enter ClrDraw as a command on a line by itself and press ENTER Be aware that zooming a graph panning left or right changing the window or changing or removing an existing graphed fun
11. some of the types of graphs it makes possible Next P ll show you Polar POL mode touching on the graph variables used for polar graphing and examples of polar functions The third and most esoteric type of graph you ll learn is Sequence SEQ mode With all four of the graphing modes your calculator supports under your belt we ll move on to drawing You can annotate any kind of graph with lines points text and sketches as you ll learn in the latter half of this chapter You can save annotated graphs to later recall Many enterprising and bored students have even used the drawing tools to doodle or draw diagrams on their calculators If you have a bunch of graphed equations that you need to use later you can save and recall those too as Pll show you GRAPH STYLES ON THE TI 84 PLUS C SILVER EDITION In chapter 3 you learned to set the color and line style of rectangular Function mode graphs The same skills apply to parametric polar and sequence graphs You can change graphs lines to any of 7 different styles and 15 different colors Let s get started with Parametric mode which is the closest of the three new graphing modes to the rectangular graphing you re now familiar with Parametric mode You ve learned all about graphing functions that map x values to y values such as Y1 2 X or Y2 sin 3X 5 One of the most noticeable shortcomings is that you can have only one y value for every x value Imagine a parabola like th
12. than the Parametric graphing in a nutshell sidebar let me lead you through two parametric graphing examples The first will show you how to graph a circle and then modify the T values plugged in to draw a semicircle The second exercise will demonstrate graphing a Lissajous curve LEASE a ju a fancy family of graphs that can only be drawn in Parametric mode Parametric example graphing a circle Graphing a circle is a challenge in Function mode but surprisingly easy in Parametric mode It s so easy because if you have an angle call it t for example and a radius call it r then the x y position of the point runits from 0 0 at angle tis simply x 7rcos t y rsin t Because parametric graphs work perfectly with equations for x and y that are both in terms of a third independent variable t you could use this to draw a circle If you pick a constant radius say r 6 and plug in every possible angle value you ll get points forming a circle around the origin 0 0 You can apply this almost exactly as written to graphing a circle in Parametric mode on your graphing calculator Enter X r 6cos T and Yir 6sin T after making sure you set your calculator to Parametric mode see the Parametric graphing in a nutshell sidebar The equations should look like the left screenshot in figure 5 4 Remember types T when you re in Parametric mode Press GRAPH and your cal culator should draw a circle as illustrat
13. with parametric graphing this time throwing a ball in two dimensions 118 5 1 2 CHAPTER 5 Expanding your graphing skills WIDOW Tm1ln 4 Thax nil LoPSe 9 a Figure 5 5 Modify the Window settings to graph a semicircle instead of a circle Press to access the menu shown at left and center where you can modify Tmin Tmax or Tstep Once you store your changes and press GRAPH you ll see the modified graph such as the semicircle at right Parametric example throwing a baseball In section 3 2 1 we used Rectangular graphing mode to look at what would happen when you threw a ball into the air You graphed time on the x axis and the height of the ball on the y axis and observed the height of the ball from the time you threw it straight upward until it fell back to the ground If you didn t throw the ball straight up but instead lobbed it across a field the problem would get more complicated Luckily because Parametric mode lets you graph x and y as a function of t you can use it to easily graph the path of a thrown baseball This example will show you how Chapter 3 taught you that the equation for the height of a thrown ball looks like this y yo vox 0 5ax In this equation a is the acceleration due to gravity 9 8 meters second yo is the initial height of the ball when thrown vo is how fast it was going in meters second when you threw it xis time and yis the height of the ball at time x in seconds If
14. 9 is the independent variable for polar graphing As with Parametric and Function Rectangular modes the Zoom menu options discussed in section 3 3 1 can be used to adjust what you see in the graph To adjust the range and granularity of 8 values plugged in to the parametric func tions use the Window menu screenshot in figure 5 11 shows Now press _y _ enter r1 7 to match the center screen shot in figure 5 11 and press to see the result You ll see what the right screenshot in figure 5 11 shows a circle of radius 7 centered on the origin Remember as we ve discussed before it looks like an ellipse even though it s a circle because your screen isn t square If you wanted to fix the confusing shape of the circle you could press and choose the 5 ZSquare option In fact for the other two examples I m going to show you in this section I recommend that you use the ZSquare setting Because polar graphing inherently deals with circular things it s particularly helpful to be able to see polar graphs in proper proportions Converting between polar and rectangular coordinates Although this section teaches you to graph polar equations you might also want to convert between polar r 0 and rectangular x y coordinates If you want to change whether the graphscreen shows polar or rectangular coordinates when you use the Trace feature you can switch between RectGC and PolarGC in the Graph Format 2na Zoom menu I
15. Min and w nMin if you have two or three sequences For example you could enter u mMin 5 Ifyou have more than one base case which happens when you have a recursive function that refers to the un term or further back you need to enter the base cases as a list For two base cases you might enter 4 2 for three 4 2 1 For our Fibonacci exercise we have two base cases enter the list 1 1 for u nMin because those are the first two terms in the Fibonacci sequence The curly braces are Jand _ as chapter 4 explained You have entered u n and u nMin so you re ready to graph the Fibonacci series by pressing GRAPH In all likelihood the graph won t look quite right An easy fix is the ZoomFit tool we looked at in the first sequence example Press zoom and choose 0 ZoomFit Much better Now you should see the plot shown on the right in figure 5 18 Ploti Plote Plots mflin l1 ee Bute lLatuce Hinin Bil 1 Man A Woohins AILT EE ZoomFit TO a ee 1234567 8 F 0 n Figure 5 18 Graphing the first 10 terms of the Fibonacci series on your calculator in Sequence mode The left shows entering the recursive function and the base cases The right is the graph after you adjust the window with ZoomFit 130 5 4 CHAPTER 5 Expanding your graphing skills VIEWING FIBONACCI SERIES VALUES You could trace along that graph but an easier way to Z 5 m examine the values of th
16. N THE GRAPHSCREEN ANNOTATING A PARABOLA The easiest way to draw on your calculator is to use its drawing tools directly on the graphscreen You can draw lines circles text points and more To select any drawing tool start on the graphscreen press to access the Draw menu and pick the tool you want The Draw menu has three tabs DRAW POINTS and STO as you can see from figure 5 20 You ll learn about the STO tab in section 5 5 On the TI 84 Plus C Silver Edition there s a fourth tab called Background but you won t need to use that unless you re writing a program When you pick a drawing tool from the Draw menu that tool will remain active until you pick another one You can draw multiple lines circles strings of text and more without needing to keep reselecting the same tool from the Draw menu You ll use that technique in this section s example annotating a parabola Start with a parabola by switching to Function mode by accessing the Mode menu entering Y 5 X 2 in the Y menu and pressing GRAPH Doesn t look like figure 5 21 No problem go to the Zoom menu and choose 6 ZStandard You can PUIHTS STU DFA E t Troran PLt Un Line Horizontal Mertical Tangent i DrawF Shade Figure 5 20 The DRAW and POINTS tabs of the Draw menu containing general tools left and point drawing tools right The STO tab contains ways to save and restore graph settings and pictures as you ll see in section 5 5
17. SAMPLE CHAPTER Using the TI 83 Plus TI 84 Plus LETTY ss Sa ss ALTIS Full coverage of the Tl 84 Plus C amet Edition zF 4 Christopher R Mitchell EE MANNING Using the TI 83 Plus TI 84 Plus by Christopher R Mitchell Chapter 5 Copyright 2013 Manning Publications brief contents PART 1 PART 2 PART 3 PART 4 BASICS AND ALGEBRA ON THE TI 83 PLUS TI 84 PLUS 1 1 m Whatcan your calculator do 3 2 a Get started with your calculator 25 3 m Basic graphing 57 4 a Variables matrices and lists 83 PRECALCULUS AND CALCULUS cccccccccccccccccccccccecs 111 5 m Expanding your graphing skills 113 6 m Precalculus and your calculator 141 7 m Calculus on the TI 83 Plus TI 84 Plus 158 STATISTICS PROBABILITY AND FINANCE cccccsccsceees 171 8 m Calculating and plotting statistics 173 9 m Working with probability and distributions 202 10 a Financial tools 223 GOING FURTHER WITH THE TI 83 PLUS TI 84 PLUS 235 11 wm Turbocharging math with programming 237 12 a The TI 84 Plus C Silver Edition 260 13 m Nowwhat 282 iil Expand your graphing skill Graphing is one of the key features of your graphing calculator chapter 3 pointed out that it s so important that it s half of the device s name In that chapter we worked with normal Cartesian graphing where you visualize equations like y x 1 In this chapter you ll see many of the advanced graphing features y
18. and pictures 137 OrawF 351nt42 ore Urawiny 351nC42 Figure 5 24 Using the DrawF DrawInv and Shade commands The left shows the three commands to enter the center shows the effect of just DrawF and DrawInv and the right is the result after the Shade command and if necessary select 1 ClrDraw from the Draw menu 2nd_ PRem Next follow these steps 1 Select 6 DrawF from the Draw menu and add 3sin X as an argument You ll end up with DrawF 3sin X as in figure 5 24 Press to execute the command 2 Choose 8 DrawInv from the Draw menu and add 3sin X again to get Draw Inv 3sin X This graphs x 3sin y from the DrawF and DrawInv commands you ll get the screenshot in the center of figure 5 24 3 Use the Shade command from the Draw menu to shade between y 3sin x 3 and y 3sin x with Shade 3sin X 3 3sin X Press ENTER and when the cal culator finishes drawing you ll see something that resembles the right side of figure 5 24 As with all the other drawing tools and in fact nearly every skill you ve learned so far I encourage you to experiment with these three new drawing features to see how they can help you with your academic work and your own drawings But what if you want to save the results of your drawing efforts In the next section you ll learn to store snap shots of the graphscreen and restore them as well as save and restore graph settings for later use Saving graph settings and pic
19. ations for the motion of the ball for a parametric plot center and a good window to see the function right 120 5 1 3 CHAPTER 5 Expanding your graphing skills NORMAL FLOAT AUTO REAL DEGREE MP Figure 5 8 Graphing the motion of a ball in two dimensions versus time on a TI 83 Plus or TI 84 Plus left or on a TI 84 Plus C Silver Edition right derivative of the expression for y Try adjusting Ymax until you get the graph to fit well or take the derivative and find the roots As a hint Ymax 5 works well You can also plug in 0 for y to figure out when the ball will hit the ground which will tell you a good Xmax Alternatively you can guess and check You ll probably end up with Xmax 70 or so as shown at the right in figure 5 7 Your resulting graph should look like one of the screenshots in figure 5 8 Because the ball started moving with rightward and upward velocity it curves upward before beginning to fall again Remember that you re now graphing as if you were standing in the bleachers watching the ball s movement You can figure out where the ball was at each instant by tracing over the graph Press trace and observe the T X and Y val ues shown Unfortunately the calculator can t find maxima minima or zeroes in Parametric mode so you ll have to use the Table or calculus to figure out when the baseball reaches the peak of its curve or when it hits the ground Let s look at a third and final pa
20. ction will also erase the graphscreen Adding a new function will not erase annotations you ve drawn as long as you don t change existing functions SKETCHING ON A BLANK SCREEN Wanta blank screen for your doodles diagrams and sketches Turn off the axes and grid by making sure AxesOff and GridOff are selected in the Graph Format menu under zoom Don t forget how to turn them on again if you need to graph but remember that graphing and mod ifying Graph Format options will erase anything you ve already drawn Using drawing tools on the homescreen It s easy to use your calculator s drawing tools on the graphscreen but sometimes you need more precision than you can get by using the tools directly on the graphscreen Or perhaps you need to draw lines or circles that are partially offscreen In this case you ll want to use the drawing tools from the homescreen Let s go through an exercise because that s the easiest way to learn anything You ll draw a circle a horizontal line and a vertical line from the homescreen Here are the steps 1 Remove any equations from the Y menu You can also turn off the axes with AxesOff from the Format menu 2nd_ zoom if you want You may want to use ZStandard and then ZSquare from the Zoom menu to make your results match my screenshots 2 Use ClrDraw from the Draw menu to start with a blank graphscreen 3 From the homescreen issue the commands Circle 2 4 10 to drawa circle centered a
21. e one in the left screenshot in figure 5 2 If you were to draw a vertical line through any x value you d hit only one point on the graph whereas a horizontal line might hit zero one or two points on the graph Rectangular mode can only create graphs like those in the left screenshot where any vertical line crosses a graphed function at one or zero points Parametric mode 115 Rectangular and Parametric Parametric only 1 1 1 0 2 2 Vertical line crosses the function in how many places Figure 5 2 The power of parametric graphing Rectangular graphing can map only one y value to each x value meaning that any vertical line will cross a graphed function in exactly one place or zero if there is a discontinuity Parametric mode can graph anything Rectangular mode can but it can also draw graphs with multiple or zero y values for each x The sideways parabola on the right side is one such example The right side of figure 5 2 shows an example of what Parametric mode can do with ease although Rectangular mode can t manage it The graph shown is essentially the equation X Y 2 which is the Y X 2 graph shown on the left turned on its side Parametric mode is only one of two ways to graph equations of the form x f y in addition to being able to graph circles functions that trace over themselves many times and much more You can see that parametric graphing is powerful but how exactly can it do what it does What gives
22. e series is to look at the table Switch into Table mode with GRAPH and you ll see values of the Fibonacci sequence side by side with their indices n as figure 5 19 illustrates You can use the 1 1 gt E B arrow keys to scroll up and down and you ll soon dis cover that you can t go to indices before n l because i gure 5 19 Viewing the there are no terms in the Fibonacci series before u 1 table of indices and values You can have negative indices in sequences on your calcu for the recursive Fibonacci lator the lowest possible n value is whatever mMin is set to i iii If you want to scroll way down or way up the table you should press and adjust the Tb1Start variable You ve learned just about everything you need to know to use Sequence graphing mode effectively on your calculator to explore recursive and nonrecursive series In fact between this chapter and chapter 3 you know all four graphing modes that your calculator offers Function Parametric Polar and Sequence modes But did you know you can draw on top of graphs and save those annotated graphs for later Drawing and saving graphs and graph settings are the subject of the remainder of this chapter Drawing on graphs Graphing calculators are wonderful math tools In a few seconds you can go from star ing at an equation in a textbook to exploring a graph of the function without having to tediously draw out the graph by hand Say you find a
23. ed on the right in figure 5 4 Parametric mode 117 Floti Fileke Floks seit decos Ti YirhesintT Figure 5 4 Graphing a circle of radius 6 in Parametric mode On the left entering the X and Y equations that define the circle as described in the text on the right the results of graphing these equations My circle isn t round Figure 5 4 shows a circle that looks more like an ellipse As men tioned in chapter 3 your calculator s default zoom sets Xmin and Ymin to 10 and Xmax and Ymax to 10 But because the screen is wider than it is tall this makes circles look stretched horizontally You can fix this for any Zoom setting by pressing and choosing 5 ZSquare If anything looks wrong and the graph doesn t look like the right side of figure 5 4 press zoom Choose the trusty 6 ZStandard option that even in Parametric mode resets graph settings to sane defaults In Parametric mode ZStandard sets the same graph edges as in Rectangular mode namely Xmin 10 Xmax 10 Ymin 10 and Ymax 10 It does something else it sets Tmin Tmax and Tstep found under wnoow To graph a pair of functions like X r and Yir your calculator plugs T values between Tmin and Tmax into the pair of functions By default Tmin 0 and Tmax 20 Your calculator can t plug in every value of T between Tmin and Tmax because there are infinitely many The Tstep value tells the calculator how much to add to T each time it plugs in
24. een only for use as a homescreen function Like Text takes pixel coordinates Row and column are pixels from the top left of the screen which is row O column O The bottom right is row 62 column 94 Pt On lt x gt lt y gt lt type Pt Off lt x gt lt y gt lt type Pt Change lt x gt lt y gt lt type Example Pt On 5 1 2 Px1 On lt row gt lt column gt Px1 Off lt row gt lt column gt Px1l Change lt row gt lt column gt Example Px1l Off 30 52 There are three more drawing commands that we haven t discussed yet in the Draw tab of the Draw menu all of which work with graphed equations You ll learn how to use those before we move on to the final topic of the chapter saving and restoring pic tures and graph settings 136 5 4 3 CHAPTER 5 Expanding your graphing skills Drawing graphlike functions DrawInv DrawF and Shade Three of the tools in the Draw menu 2nd_ PRem take functions as arguments and draw something based on equations All three work only in Function mode or act as if the calculator is in Function mode regardless of the current settings These three functions are as follows DrawInv Draws the inverse of a Y rectangular equation Useful for drawing X equations see the Graphing X equations sidebar for more information This function takes one argument the equation to graph the inverse of For example DrawInv X 3 DrawF
25. ents will use in their academic years and beyond Samuel Gockel University of Illinois at Urbana Champaign The perfect complement to math and statistics courses of all levels Ryan Boyd University of Texas at Austin This is THE manual for your calculator Jonathan Walker computer science student NDSU ISBN 13 976 1 b617290 44 4 gt 1 617 84 X 290 290848 ISBN 10 781617
26. errific tutorials that guide you through the most important techniques dozens of examples and exercises that let you learn by doing and well designed reference materials so you can find the answers to your ques tions fast Using the TI 83 Plus TI 84 Plus starts by making you com fortable with these powerful calculators screens buttons and special vocabulary Then you ll explore key features while you tackle problems just like the ones you ll see in your math and sciences classes What s Inside e Get up and running with your calculator fast e Lots of examples e Special tips for SAT and ACT math e Covers the color screen TI 84 Plus C Silver Edition Written for anyone who wants to use the TI 83 Plus TI 84 Plus No advanced knowledge of math and science required Christopher Mitchell is a teacher PhD candidate and recog nized leader in the calculator enthusiast community You ll find Christopher aka Kerm Martian and his cadre of calculator experts answering questions and sharing advice on his website cemetech net To download their free eBook in PDF ePub and Kindle formats owners of this book should visit manning com UsingtheTl 83Plus Tl 84Plus MANNING US 24 99 Can 26 99 The user manual but shorter clearer and much more entertaining Louis Becquey Joseph Fourier University Grenoble Expertly captivates readers at any level of study and provides practical examples that stud
27. f each term As it often does your calculator has your needs covered There are several ways you can get values but two are particularly fast and easy a Use the Trace feature Enter one or more sequences in the Y menu graph them then press Trace Use 4 and P to move between sequence terms the and 4 keys switch between different sequences if you entered more than one Chapter 3 explains more about using Y a Use the Table tool In section 3 4 you learned how to use the Table tool to see function rectangular graph values The Table tool works for all four graphing modes and is useful for sequences Graph one or more sequences and then press GRAPH Figure 5 17 shows what the Table tool looks like for the sequence of squares you experimented with in this section Because the TI 84 Plus C Silver Edition has a larger screen you may want to switch to the Graph Table split screen mode from the Mode menu and then view the table as illus trated on the right of figure 5 17 Remember that if you want to change where the table starts or the spacing between values in the table you can adjust Tb1Start and ATbl1 from the TblSetup menu accessed with window The sequence of squares I showed you in this section was a relatively easy and nonrecursive sequence You d be missing an important piece of your sequence graphing knowledge if you didn t also go through a recursion exercise Sequence example the Fibonacci series T
28. f you have numerical r or x y coordinates you can convert them to the opposite form on the homescreen using four functions from the Angle menu All will heed your current Radian Degree setting in the Mode menu Press and use one of these 5 2 1 Graphing polar functions 123 To help you get a feel for your calculator s polar graphing features I want to take you through two examples The first will show you how to graph a spiral and the second will introduce a graph called a polar rose Let s begin with a spiral a surprisingly easy shape to graph in Polar mode Polar example a spiral Graphing a spiral might seem rather impossible at first but Polar mode makes it sim ple If you start with a circle which has a constant radius then to make a spiral you can break the circle and pull one end into the center A first attempt at a spiral might be r 0 so that when 9 0 r 0 when 0 7 r 7 and when 0 27 r 27 In other words as 9 increases the radius r increases as well The initial r 0 spiral gets very far from the origin very quickly You can make it a tighter spiral by dividing 0 by a constant say n giving you ri 0 m The left screenshot in figure 5 12 shows entering this equation and the right screenshot illustrates what happens when you press the key note that this screenshot was taken with ZStandard rather than ZSquare That s not much of a spiral How can you make it continue Remember that by
29. hat acceleration due to gravity called either g or a is 9 8 m s negative because it s acceleration downward To make things easy let s assume that the ball starts at horizontal position xo 0 meters and that if the person throwing the baseball is about 2 meters tall the ball s height when he releases it is yo 1 8m x x0 Voxt 0 39 39t 39 39t y yo vot 0 5af 1 8 6 95t 4 97 Enter these equations as Xir and Yir Press _Y and if there are any equations already entered clear them or disable them If you don t see pairs of parametric equa tion prompts as at center in figure 5 7 go to the Mode menu and make sure you re in Polar mode DEGREE SYMBOL Figure 5 7 shows the degree symbol If you want to use sin and cos without specifying an angle mode in the Mode menu you can put the degree or radian symbol after a number to specify whether it s a degree or radian angle Both symbols are in the Angle menu _2nd_ APPS You also need to choose a window Because the ball starts at x 0 Xmin 0 seems like a good choice and because the ball can t go below the ground Ymin 0 is also logical You can find the maximum y height the ball will reach graphically or by taking the 4c Floti Flotz Flot WT HOO osi lH i 39 392316012 s11 B39 39T 4Hsint1y a le 6 945927187 Figure 5 7 Calculating the x and y components of the baseball s initial velocity left Entering the equ
30. here are endless examples of recursive functions that are useful in programming science and engineering but the Fibonacci series is a classic and easy to understand example The Fibonacci series was invented by a 13th century Italian mathematician 128 CHAPTER 5 Expanding your graphing skills NORMAL FLOAT AUTO REAL RADIAN MP T 0 1 2 3 4 5 6 8 9 1 Figure 5 17 Checking sequence values with the Table tool After graphing a sequence press to view the table As with normal graphing you can modify Table tool settings from the TbISetup _2nd_ winvow menu If you use a TI 84 Plus C Silver Edition and switch to the Graph Table mode from the Mode menu you ll see the results on the right side instead named Leonardo Fibonacci It s a sequence where the first two terms are 1 and every subsequent term is the sum of the two terms right before it The base cases see the More on recursion sidebar of the Fibonacci sequence are wi 1 and w 1 and the recursive definition iS Un Un 1 Un 2 e uw wtm l l 2 me Wm4 uw w 2 1 3 mE Ww m uwy 34 2 5 wo ug us We can t tell without calculating w and us which requires u7 and we In testing the Fibonacci sequence on your calculator you ll learn how to enter the expression for a recursive series including teaching your calculator the base case s We ll explore how changing the base cases changes every other term in the series I ll also reiterate the
31. it its power Your favorite math book or teacher can give you more details but the essentials are that every point in a parametric graph is defined by two functions not just one In rectangular graphing as shown on the left in figure 5 3 you take each x value and plug it into a single function written as y f x and the y value pops out By definition each x value can have only a single y value because a function can t give two different y values when the same x value is plugged in 116 5 1 1 CHAPTER 5 Expanding your graphing skills DEPENDENT DEPENDENT VARIABLE VARIABLE gt DEPENDENT VARIABLE INDEPENDENT VARIABLE Rectangular Parametric Figure 5 3 The mechanical differences between rectangular and parametric graphing With rectangular graphing every y value is generated by passing a corresponding x value through the function to be graphed Parametric graphing is much more powerful because graphed functions can pass through the same x coordinates y coordinates and even x y point multiple times Parametric graphing by contrast defines every point on the graph by two functions call them x f t and y g t By making both x and y depend on but not on each other you can express many more graphs You can have graphs with multiple x values for the same y multiple y values for the same x and even graphs that intersect with themselves or trace over themselves If you have time for a longer introduction
32. ition you should instead refer to table 12 2 in section 12 3 Line Draws a line between two points Line lt xi gt lt Yyi gt lt X2 gt lt Y2 gt On the graphscreen move the cursor Example Line 1 0 8 2 8 to the first point press ENTER move it to the second point and press again Horizontal Draws a horizontal line across the whole Horizontal lt y coordinate gt screen Move the line to where you want it Example Horizontal 1 5 and press to make it permanent Vertical Draws a vertical line down the whole Vertical lt x coordinates screen Move the line to where you Example Vertical 4 want it and press to make it permanent Tangent Draws a line tangent to a given function ata Tangent lt func gt lt x coord gt given x point Always assumes that functions Example Tangent 3X 1 25 are rectangular FUNC functions even if the calculator isn t in Function mode On the graphscreen you select the function and then the point Circle Draws a circle with a center and radius On Circle lt centers the graphscreen select the center press lt center y gt lt radius gt ENTER and then choose a point on the edge Example Circle 2 71 4 of the circle Drawing on graphs 135 Table 5 1 Drawing functions in the Draw tab of the Draw menu You must first go to the graphscreen press 2nd and select the function you want and then use it You can also go to the homescreen in
33. n interesting point in such a graph You want to circle the point add some text next to it to explain what it is and save the graph to show to your teacher later At this point you ll probably end up try ing to sketch the graph on a page in a notebook and annotate it by hand After you read this section though you ll know how to draw those annotations directly on the graph By the end of section 5 5 you ll also have learned how to save the annotated graph in your calculator s memory Your calculator includes a host of drawing tools including 10 ways to draw lines circles text and even pieces of functions and 6 ways to draw points You can draw directly on the graphscreen or you can run drawing functions as homescreen func tions In this section Ill show you how to draw on top of graphs You ll learn to use these functions directly on the graph and which ones can also be used on the home screen I ll also show you the three graphing tools that are related to graphed func tions DrawInv DrawF and Shade By the time you get to the end of this section you ll be well versed in drawing on your calculator And because I m nothing if not a realist I know that many of you will end up using these tools to sketch drawings of your own I did it too back in the day Let s start with drawing functions like Horizontal Line and Text used on the graphscreen and homescreen 5 4 1 Drawing on graphs 131 Graphscreen drawing
34. of the ellipse to the top or bottom will be equal to a and the semi major radius the distance from the middle of the ellipse to the left or right side will be equal to b IMPROVING PARAMETRIC RESOLUTION Want to make that curve look smoother You can apply a lesson I taught you in the parametric cir cle example make Tstep smaller Try changing Tstep to 1 48 by pressing wnoow changing Tstep and then pressing GRAPH The curve should appear much smoother as in figure 5 10 although it will take longer to render Parametric mode is a powerful change of pace great for expressing functions that are impossible in Rectangular mode We ll now move on to Polar Figure 5 10 A Lissajous curve graphed mode which offers some secrets of its own on a TI 84 Plus C Silver Edition with Tstep 7 48 Graphing polar functions Polar mode shares some attributes of both function and parametric graphing As in Function mode single equations map an independent variable to a dependent vari able Like Parametric mode Polar mode gives you a way to express graphs that would be impossible in Rectangular mode and that overlap themselves In Polar mode the independent variable is O the Greek letter theta which repre sents an angle By default O ranges from 0 to 2m The dependent variable is r or radius a distance from the origin at angle 0 There are six polar equations numbered r through rs Once you switch your calculator into Polar mode
35. on No u nMin is necessary because the equation isn t recursive The center is the result with a ZStandard window and the right is the result with Ymin adjusted to 0 and Ymax changed to 100 5 3 2 Graphing sequences 127 USING ZOOMFIT WITH SEQUENCES The Sequence graphing in a nutshell sidebar said that J ZoomFit is handy for sequences so let s give that a try ae Whether or not you already adjusted the window doesn t mat Xmin 1 l ter go to the Zoom menu and choose 0 ZoomFit the tenth Xmaxz10 item in the list Your calculator will look at the values of nMin y Ymia l and nMax plus the values of the sequence at the u n v n and or w n and tailor a good window to the sequence s Figure 5 16 Using you ve graphed ZoomFit in the Zoom Try it for this sequence u n n With the graph entered menu to automatically pick a good window to see part choose ZoomFit from the Zoom menu and you ll immedi of a sequence ately see the result which should look like figure 5 16 If nMax is set to a large value this might take a bit longer The calculator decided to set Xmin mMin 1 Xmax nMax 10 Ymin u 1 1 and Ymax u 10 100 This does a good job of showing all the relevant terms in the sequence between n l andn 10 EXAMINING SEQUENCE TERMS TRACING AND THE TABLE TOOL It s great to visualize sequences on the graphscreen but often you also want to know exact numerical values o
36. ook at your calculator s picture variables and features Saving and recalling picture variables Just as your calculator has numerical variables for storing numbers list variables for storing lists and matrix variables for storing matrices it has a set of 10 picture vari ables for storing pictures or snapshots of the graphscreen Named Pic1 through Pic9 plus Pic0 each is a location in your calculator s memory and each can store a full image of the graphscreen WHY NOT PICO THROUGH PIC9 In section 5 5 1 I refer to Pic1 through Pic9 plus Pico instead of just saying Pico through Pic9 This isn t an error or an attempt to confuse you Because your calculator thinks of PicO as Pic10 it s the last of the 10 picture variables after Pic9 The StorePic and RecallPic commands which respectively store and recall picture variables i a Pictures can be found in the STO tab of the Draw menu Foe aGDE To access this menu shown in figure 5 25 press Fecal 16DE Vorri graph settings gt gt Once you choose StorePic or RecallPic the command is pasted to the homescreen Add a 1 2 3 9 or 0 after either Figure 5 25 The STO tab of the Draw menu from which you can save ture number and recall both pictures and graph settings GDBs StorePic or RecallPic to save or open that pic PICTURE CAVEATS If you StorePic a picture that already exists your calculator will overwrite the old pic
37. ored examples of polar functions from circles and spirals to rose shaped graphs We worked through sequence graphing exercises of a recursive series like the Fibo nacci series and a nonrecursive series like the squares of the positive integers In the latter half of the chapter I showed you the many drawing and annotation tools that your calculator offers From ways to mark up graphed equations to shade functions and graph X equations there s not much you can t draw on your calcula tor s graphscreen You even learned how you can use the graphscreen as a creative canvas for freehand sketches and diagrams We concluded with a look at the tools for storing and recalling pictures which store the pixel by pixel contents of the graph screen and GDBs which hold the currently graphed equations and graph settings As we move forward into the rest of part 2 of this book you ll be graduating to the realm of precalculus and calculus Chapter 6 focuses on precalculus features of your graphing calculator that you haven t explored yet You ll look at complex numbers the nitty gritty of trigonometry limits and logarithms Onward MATHEMATICS GENERAL Using the TI 83 Plus TI 84 Plus Full Coverage of the TI 84 Plus C Silver Edition Christopher R Mitchell ith so many features and functions the TI 83 Plus TI 84 Plus graphing calculators can be a little in timidating This book turns the tables and puts you in control In it you ll find t
38. our calculator includes such as new graphing modes and annotating graphs with drawings Why would you want to graph in other schemes than Cartesian graphing any way The short answer is that there are many functions and graphs that you can t represent in the form y f x which defines the vertical position y of each point in the graph by passing its horizontal position x through a function Figure 5 1 shows the four different types of graph coordinate systems that your calculator can work with Whereas chapter 3 taught you rectangular or Cartesian graphing this chapter will introduce the other three modes in figure 5 1 We ll first go through Parametric 113 114 5 1 CHAPTER 5 Expanding your graphing skills Graph Coordinate Systems y x f y t Oea unt x a ri Se a Cartesian Parametric Polar Sequence Figure 5 1 The four types of graph coordinate systems your calculator knows At far left Cartesian or rectangular coordinates where x is the independent variable and y x is the dependent variable Parametric mode at center left makes both x and y dependent on a third independent variable t which lets you graph functions that can t be expressed as Cartesian functions in the form y x At center right polar coordinates where the radius r is dependent on the independent variable 0 angle Finally sequence graphing is for normal or recursive functions applied to independent n values PAR mode including
39. quation for a polar rose is r 0 a sin 0 The a the annotated points on the a number of petals For the sake of this discussion the gen term controls the size of the rose and is roughly equivalent graph showing the 9 r coordinates at several points to a radius from the middle of the rose to the tip of any along the spiral petal The b term controls how many petals the rose has If bis even any rose graphed from r 0 a sin b0 has 2b petals if b 2 it has 4 petals and if b 5 it has 10 petals a If bis odd the rose has b petals if b 3 it has 3 petals and if b 7 it has 7 petals Pll explain how you can graph polar roses but it s up to you to decide how many pet als you want A few of the possible roses with even numbers of petals are shown in fig ure 5 14 First pick your number of petals Use the preceding rules about odd and even val ues of b to decide what value b needs to have I recommend a radius of a 8 but you re welcome to choose any radius you want For the following discussion PI assume values of a 8 and b 4 Here s how you graph the 7 8sin 40 rose 1 Make sure you re in Polar mode If necessary choose ZStandard and then ZSquare from the Zoom menu to set the edges of the graph to good values 2 Press _Y clear out any existing equations and enter r 8sin 40 The easiest way to type the 8 variable is to press the key as long as you re in Polar mode 3 Press GRAPH and
40. rametric graphing example a fancy family of curves called Lissajous curves Parametric example a Lissajous curve A Lissajous curve also called a Bowditch curve is a family of parametric functions cre ated with sine and cosine Figure 5 9 shows an example of a Lissajous curve drawn on a graphing calculator You can test it yourself by plugging in the two equations shown on the left in figure 5 9 resulting in the graph shown on the right As always press zoom and choose 6 ZStandard if you don t get the same graph To give you a bit of background a Lissajous curve is any of a family of parametric curves of the form x t asin ct d y t b sin et Ploti Flotz Flot Sai Thesintolr iy aerostols Figure 5 9 Graphing a Lissajous curve in Parametric mode The left screenshot shows the equations to enter and the right shows the result 5 2 Graphing polar functions 121 In these equations a b c d and eare constant numbers not variables In the example I just showed you we picked a b 8 c 5 d 0 and e 3 You should try fiddling with these variables to get different curves in the family some of which are unique The keen eye might also notice that circles are technically in the Lissajous curve family If you let c e 1 d 1 2 and a b you ll get a circle of radius a You can also graph an ellipse by keeping c e 1 and d 1 2 but making a and b unequal The semi minor radius the distance from the middle
41. reen but the equations that you have currently graphed plus the window and format settings Then you should use a GDB Saving and recalling graph databases A graph database or GDB is a somewhat different animal than a picture variable There are 10 of them GDB1 through GDB9 plus GDBO and they re located in your calcu lator s memory But instead of holding an exact snapshot of the pixels on the graph screen GDBs hold all the following Any equations you have entered in the Y menu for all four modes The current graph mode The Graph Format settings from the Format menu including whether the axes the grid and coordinates are on or off The window variables including Xmin Xmax Ymin Ymax and any other values in WwinDow m On the TI 84 Plus C Silver Edition it also saves the colors of each graphed equa tion the axes color the grid color and the color of the border around the graph It does not save the background picture or color Saving and recalling GDBs use the StoreGDB and Recal1GDB functions from the STO tab of the Draw menu as shown in figure 5 25 You follow StoreGDB and Recal1GDB with a number from 0 to 9 exactly as with StorePic and RecallPic I showed you how to list the picture variables on your calculator and archive and delete them from the Memory menu You can list delete and archive the GDB variables nearly the exact same way press to access the Memory menu then choose 2 Mem Mgmt Del
42. stead to use many of these functions If you have a TI 84 Plus C Silver Edition you should instead refer to table 12 2 in section 12 3 continued Text Writes out text Move the cursor where you want the top left corner of the text to be and start typing no need to press ENTER Remember to press if you want to type letters For entering Text on the homescreen row and column are pixels from the top left of the screen which is row O column O The bottom right is row 62 column 94 Somewhat like drawing with an Etch a Sketch Move the cursor to where you want to start press to put the pen down and move the pen to draw a black line behind it You can press to lift the pen and move it to a new spot to draw again Text lt row gt lt columns gt STRING Example Text 57 1 GRAPH TITLE You can t use the Pen as a home screen command Table 5 2 Point and pixel commands all in the Points tab of the Draw menu Pt On Pt OfE Pt Change Px1 On Pxl Off Px1 Change Once you enable this tool you can freely move the cursor around the graphscreen Every time you press ENTER it will turn the point on to black turn it off to white or change it From the homescreen this com mand uses x y coordinates If you omit lt type gt it draws a single dot You can enter 2 for type for a square and 3 for a cross Can t be drawn on the graphscr
43. t 2 4 of radius 10 you can see the command in figure 5 22 or figure 5 23 If you re using a TI 84 Plus C Silver Edition use the command Circle 2 4 10 RED you can find RED under gt gt 4 Use the Horizontal 6 and Vertical 8 options to draw two lines tangent to the circle You should see something that matches the right side in figure 5 22 or figure 5 23 If you have a TI 84 Plus C Silver Edition try Vertical 8 GREEN instead of Vertical 8 Figure 5 22 Using drawing commands from the homescreen to draw a circle a horizontal line and a vertical line For this example I turned off the axes with AxesOff and used ZStandard and ZSquare to set up the window 134 CHAPTER 5 Expanding your graphing skills NORMAL FLOAT AUTO REAL RADIAN MP NORMAL FLOAT AUTO REAL RADIAN MP Vertical 8 GREEN Figure 5 23 The same set of drawing commands as figure 5 22 on the TI 84 Plus C Silver Edition used colors in two of the commands found in the Color tab of the Vars menu For the full list of drawing commands look at tables 5 1 and 5 2 Table 5 1 contains most of the interesting commands from the Draw menu and table 5 2 lists the com mands from the Points tab Table 5 1 Drawing functions in the Draw tab of the Draw menu You must first go to the graphscreen press and select the function you want and then use it You can also go to the homescreen instead to use many of these functions If you have a TI 84 Plus C Silver Ed
44. t starts 1 1 2 3 5 8 13 21 In this section lll demonstrate graphing the Fibonacci sequence on your calculator As in the previous discussions of new graph modes the Sequence graphing in a nut shell sidebar tells you everything you need to know to quickly get started graphing sequences If you have time to explore two exercises with me Pll show you how to graph and examine two sequences The first will be the sequence un n where each term is the square of its index The second sequence is the Fibonacci sequence a clas sic and fun example Sequence graphing in a nutshell To switch to Sequence mode press MODE move the cursor to SEQ and press ENTER m When you go to the Y menu you ll see one setting nMin and three pairs of items u n v n and w n are the sequences to be graphed and u mMin v nMin and w nMin are lists of initial values necessary for recursive sequences When you press the key in SEQ it types an n instead of an X because n is the independent variable for sequence graphing To type recursive functions you might need to type u n 1 u n 2 v n 1 and similar expressions The sequence equation letters u v and w can be typed with ea 8 and 9 respectively As for every other mode the Zoom menu options discussed in section 3 3 1 can be used to adjust what you see in the graph ZoomFit is particularly useful for sequences a The Window menu for Sequence mode
45. tools No matter what you re trying to draw on your calculator from graph annotation to geometry diagrams to doodles your calculator has a full complement of tools to help you The best way to learn to use the drawing tools is to play around with them so I m going to cover the high level of using the tools in general and let you experiment for yourself To guide you I ve created tables 5 1 and 5 2 later in this section explaining each of the tools With few exceptions most drawing tools can be used one of two ways Go to the graphscreen by pressing GRAPH Optionally you can graph equation s first From the graphscreen press to get to the Draw menu pick a tool and use it Continue until you re happy with your creation Go to the homescreen and enter drawing commands as functions like round and gcd and all their friends This requires memorizing the arguments to each function the values you put in parentheses after the function names Chapter 2 explained how to use functions on the homescreen Pll start you with drawing by example namely graphing a parabola and annotating that graph We ll then touch briefly on homescreen drawing commands and conclude this section with a table of the drawing tools you ll use most frequently If you have a TI 84 Plus C Silver Edition you may also want to refer to section 12 3 which discusses the differences between drawing on the older and newer calculators USING DRAWING TOOLS O
46. ture without warn ing you If you try to RecallPic a picture that doesn t exist your calculator will show an ERR UNDEFINED message If you want to see a list of which picture variables you have on your calculator you can press to access the Memory menu then choose 2 Mem Mgmt Del and choose 8 Pic You ll see a list of any picture variables on your calculator and you can archive unarchive and delete pictures from here Press next to any picture to delete it which saves memory RAM If you archive a picture by pressing next to it which adds an asterisk then it won t take up RAM and will survive a RAM clear But you can t StorePic or RecallPic that variable until you unarchive it by returning to the Memory menu and pressing next to it again To demonstrate saving and recalling pictures I drew a sketch on the graphscreen as shown on the left in figure 5 26 Next I cleared the graphscreen with ClrDraw and then recalled the picture that I saved as shown in the center in figure 5 26 When I pressed GRAPH the sketch I had drawn was restored 5 5 2 TI 4 _ Saving graph settings and pictures 139 Figure 5 26 Demonstrating how StorePic and RecallPic let you save a snapshot of graphs annotations or drawings on the graphscreen Here store the DNA doodle I drew clear the screen erasing the doodle and then use RecallPic to restore the sketch What if you want to save not the exact contents of the graphsc
47. tures The final skills of this chapter are among the easiest of the new graphing and drawing skills and thus I saved them until the end to relax your brain after the rigors of graph ing and drawing In addition there s no point saving pictures of the graphscreen and your graph settings if you have neither graphs nor pictures In a nutshell this section will show you how to take a snapshot or picture or screenshot if you prefer of the graphscreen and save it into your calculator s memory You can later restore one of those pictures to return the graphscreen to how it looked when you took the picture regardless of any new graphs drawings mode changes and format changes that you made in the meantime This section will also explain graph databases GDBs a way to store all currently graphed equations plus your Graph Format settings and graph win dow into memory You can then later restore a GDB to return your graphs and settings to the way they were when you stored the GDB Saving and recalling pictures pics is useful whenever you ve made an annotated graph or a drawing that you want to store for later Pictures are the first topic we ll 138 5 5 1 CHAPTER 5 Expanding your graphing skills examine The second is saving and recalling GDBs which is handy when you have your graphed functions and graph format settings exactly as you want them and want to be able to later switch back to those settings Let s get right to it with a l
48. you switch to the more intuitive variable for time you can use x for the horizontal posi tion at time tand make xo represent the starting horizontal position of the ball If you threw a ball across a field with an initial velocity vw here s what the two equations describing its motion would look like X X0 Voxt y yo vot 0 5at GRAPHING A BASEBALL S PATH You can use these equations to graph the path of a thrown baseball over time A pro fessional baseball player might be able to throw a ball at 90 miles per hour or about 40m s Let s say he throws it at a 10 angle to the ground as shown in figure 5 6 Phys ics and trigonometry tell us that the xcomponent and ycomponent of that initial velocity shown in figure 5 6 are Vox vo cos 10 40 0 985 39 39 m s voy vo sin 10 40 0 174 6 95 m s Parametric mode 119 Yoy H0Osin 10 6 95 m s i 7 40 m s 10 gt ZO TE gt ox HO cos 10 39 3 m s N Figure 5 6 Throwing a baseball at a 10 degree angle to the ground and seeing how to calculate the x and y components of the initial velocity of 40 m s The math to calculate these components is shown at left in figure 5 7 along with the equations you Il soon enter You can create a pair of parametric equations describing the motion of the base ball in figure 5 6 by plugging vox and voy for this ball into the equations for x and y and substituting xo yo and aas well You may recall t
49. your rose will appear As in previous exercises I encourage you to play around with different graph settings and values for the rose equation if you have time Tweak a and b to see how the rose changes and try varying Ostep to make the rose petals smoother or rougher Now you ve worked with two new graphing modes in this chapter and you know how to use three of the four modes your calculator offers The fourth and final mode Sequence is the least used but still powerful and handy Figure 5 14 Three sample polar roses all of which have a 8 From left to right b 2 b 4 and b 6 5 3 Graphing sequences 125 Graphing sequences Sequences are special types of equations that produce successive terms rather than a continuous set of values Each term of a sequence is a number and it may or may not depend on previous values in the sequence Here are two simple sequence examples A simple sequence where terms did not depend on the values of previous terms would be the set of positive even numbers 2 4 6 8 10 12 If you want to name terms of the sequence you can call the first one w the second one u and so on The nth term is un so the equation for this sequence is Un 2n for example uw 2 and us 6 A simple sequence where terms depend on the values of previous terms would be a sequence where each term is the sum of the previous two terms This is a special sequence called a Fibonacci sequence and i
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