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# Evaluating functions with ti 83 84

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## Evaluating functions with ti 83 84Document Transcript

• TC3 → Stan Brown → TI-83/84/89 → Evaluating Functions revised 5 Aug 2007 Evaluating Functions with TI-83/84 Copyright © 2001–2012 by Stan Brown, Oak Road SystemsSummary: Very often, you need to evaluate a function for several values of x. Examples: inding a limit or an approximation numerically, plotting on paper by plotting points. This page gives you two techniques, creating a table of values and tracing a graph. We’ll use this function to show the methods: f(x) = (x³ − 1) / (x − 1).Contents: Both Methods: Enter the Function Method 1: Table of Values Method 2: Trace on the GraphSee also: Graphing Functions on TI‐83/84 Graphing Piecewise Functions on TI‐83/84Both Methods: Enter the FunctionYou have to tell the TI‐83/84 what your function is, before you can use Method 1 or Method 2.If any other plots are active Press [Y=].(highlighted), you need to turn them off.If nothing is highlighted on your Y= Use the blue arrow keys to move the cursor onto everyscreen, you can skip this step. highlighted = sign or Plot number, and press [ENTER]. Your screen will look different, but that’s okay as long as you haveIt so happens that I had Plot1 and Y2 nothing highlighted.active, as you can see below left. In theright‐hand panel I’ve deactivated bothof them. 1 of 4
• Enter the function. Cursor to any empty Y= slot. Press [(] [x,T,θ,n] [^] [3] [−] 1 [)] [÷] [(] [x,T,θ,n] [−] 1 [)] [ENTER].You may be using a different function number, but make sure your entrylooks like my Y3. If you made a mistake, go back and edit it.Method 1: Table of ValuesThis method works strictly by numbers. Press [2nd WINDOW makes TBLSET].The irst two rows don’t matter. Press [▼] [▼] to get to the “Indpnt” row. This controls the independent variable x. Press [►] [ENTER] to select “Ask.” Press [▼] [ENTER] to select “Depend: Auto.”Your top two rows may be different, but your bottom two rows will looklike the screen at right. These settings tell the TI‐83/84 to ask you forvalues of the independent variable x, then automatically calculate thevalues of the dependent variable f(x).Now you can evaluate the function at Press [2nd GRAPH makes TABLE].selected x values. You may see some values on the table screen. They don’t do any harm, but if you want you can get rid of them by hitting [DEL] several times.Enter the x values, one at a time. For Enter each x number and press [ENTER]. The TI‐83/84instance, to home in on the limit as x immediately displays the function value.approaches 1, we might enter .5, .75, .9,.99, and so on. 2 of 4
• Method 2: Trace on the GraphThis method is a little more work, but you get a picture of the function. Start by entering the function onthe Y= screen as shown above.Set formatting so that the desired Press [2nd ZOOM makes FORMAT].information will appear on your graph. The other settings are not critical, but you need CoordOn and ExprOn. Press the arrow keys and [ENTER] to set the modes.Either set up the Window screen, or use Press [ZOOM] [6] to select ZoomStd. The graph should appear“Zoom Standard” for a irst look at the (below left).graph. It happens that we want the limitas x goes to 1. Since that its within the Press [TRACE], then an x value, then [ENTER]. You should seestandard window, we’ll use Zoom the function, the x value, and the y value displayed (belowStandard this time. right).Enter any other values, such as .99, .995, You don’t have to press [TRACE] again. Simply enter each new.999. The dot will move along the graph, x value, followed by [ENTER].and the new y values will be displayed. 3 of 4
• For this function it’s not an issue, but for other functions if your x value is outside the window, you needto press [WINDOW] and adjust Xmin or Xmax. You can only trace x values that are between Xmin andXmax.This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/ 4 of 4
• TC3 → Stan Brown → TI-83/84/89 → Graphing Functions revised 5 Aug 2007 Graphing Functions on TI-83/84 Copyright © 2001–2012 by Stan Brown, Oak Road SystemsSummary: It’s pretty easy to produce some kind of graph on the TI‐83/84 for a given function. This page helps you with the tricks that might not be obvious. You’ll be able to ind asymptotes, intercepts, intersections, roots, and so on.See also: Evaluating Functions with TI‐83/84 Graphing Piecewise Functions on TI‐83/84Contents: Graphing Your Function Common Problems Tuning Your Graph Zooming Adjusting the Window Adjusting the Grid Exploring Your Graph Domain and Asymptotes Function Values Intercepts Multiple Functions IntersectionThe techniques in this note will work with any function, but for purposes of illustration, we’ll useGraphing Your FunctionStep 1: Clear unwanted plots.You need to look for any previously set Press [Y=] (the top left button). Look at the top of the screen.plots that might interfere with your If any of Plot1 Plot2 Plot3new one. is highlighted, cursor to it and press [ENTER] to deactivate it. (No information is lost; you can always go back and reactivate any plot.) To verify 1 of 7
• that you have deactivated the plot, cursor away from it and check that it’s not highlighted.(Sometimes you might want to graph Now check the lines starting with Y1=, Y2=, and so on. If any =more than one function on the same sign is highlighted, either delete the whole equation oraxis. In this case, make sure to deactivate it but leave it in memory. To delete an equation,deactivate all the functions you don’t cursor to it and press the [CLEAR] button. To deactivate itwant to graph.) without deleting it, cursor to its = sign and press [ENTER]. My screen looked like this after I deactivated all old plots and functions.Step 2: Enter the function.If your function is not already in y= Cursor to one of the Y= lines, press [CLEAR] if necessary, andform, use algebra to transform it before enter the function.proceeding.Two cautions: For x, use the [x,T,θ,n] key, not the [×] (times) key. The TI‐83/84 follows the standard order of operations. If there are operations on top or bottom of a fraction, you must use parentheses — for x+2 divided by x−3, you can’t just enter “x+2/x−3”.Check your function and correct any Use the [◄] key and overtype any mistakes.mistakes. To delete any extra characters, press [DEL].For example, if you see a star * in placeof an X, you accidentally used the times If you need to insert characters, locate yellow INS above thekey instead of [x,T,θ,n]. [DEL] key. Press [2nd DEL makes INS] and type the additional characters. As soon as you use a cursor key, the TI‐83/84 goes back to overtype mode.Step 3: Display the graph. 2 of 7
• “Zoom Standard” is usually a good Press [ZOOM] [6].starting point. It selects standardparameters of ‐10 to +10 for x and y.Common ProblemsIf you don’t see your function graph anywhere, your window is probably restricted to a region of the xyplane the graph just doesn’t happen to go through. Depending on the function, one of these techniqueswill work: ZoomFit is a good irst try. Press [ZOOM] [0]. (Thanks to Marilyn Webb for this suggestion.) You can try to zoom out (like going higher to see more of the xy plane) by pressing [ZOOM] [3] [ENTER]. Finally, you can directly adjust the window to select a speci ic region.For other problems, please see TI‐83/84 Troubleshooting.Tuning Your GraphYou can make lots of adjustments to improve your view of the function graph. ZoomingThe window is your ield of view into the xy plane, and there are two main ways to adjust it. Thissection talks about zooming, which is easy and covers most situations. The next section talks aboutmanually adjusting the window parameters for complete lexibility. Here’s a summary of the zooming techniques you’re likely to use: You’ve already met standard zoom, which is [ZOOM] [6]. It’s a good starting point for most graphs. 3 of 7
• You’ve also met zoom it, which is [ZOOM] [0]. It slides the view ield up or down to bring the function graph into view, and it may also stretch or shrink the graph vertically. To zoom out, getting a larger ield of view with less detail, press [ZOOM] [3] [ENTER]. You’ll see the graph again, with a blinking zoom cursor. You can press [ENTER] again to zoom out even further. To zoom in, focusing in on a part of the graph with more detail, press [ZOOM] [2] but don’t press [ENTER] yet. The graph redisplays with a blinking zoom cursor in the middle of the screen. Use the arrow keys to move the zoom cursor to the part of the graph you want to focus on, and then press [ENTER]. After the graph redisplays, you still have a blinking zoom cursor and you can move it again and press [ENTER] for even more detail. Your viewing window is rectangular, not square. When your x and y axes have the same numerical settings the graph is actually stretched by 50% horizontally. If you want a plot where the x and y axes are to the same scale, press [ZOOM] [5] for square zoom.There are still more variations on zooming. Some long winter evening, you can read about them in themanual. Adjusting the WindowYou may want to adjust the window parameters to see more of the graph, to focus in on just one part, orto get more or fewer tick marks. If so, press [WINDOW]. Xmin and Xmax are the left and right edges of the window. Xscl controls the spacing of tick marks on the x axis. For instance, Xscl=2 puts tick marks every 2 units on the x axis. A bigger Xscl spaces the tick marks farther apart, and a smaller Xscl places them closer together. Ymin and Ymax are the bottom and top edges of the window. Yscl spaces the tick marks on the y axis.If you want to blow up a part of the graph for a more detailed view, increase Xmin or Ymin or both, orreduce Xmax or Ymax. Then press [GRAPH]. If you want to see more of the xy plane, compressed to a smaller scale, reduce Xmin and/or Ymin, orincrease Xmax or Ymax. Then press [GRAPH].Many of the graph windows shown in your textbook will have small numbers printed at the four edges.If you want to make your graphing window look like the one in the textbook, press use the numbers atleft and right edges for Xmin and Xmax, the number at the bottom edge for Ymin, and the number at thetop edge for Ymax. Adjusting the GridThe grid is the dots over the whole window that line up to the tick marks on the axes, kind of like graph 4 of 7
• paper. The grid helps you see the coordinates of points on the graph. If you see a lot of horizontal lines running across the graph, your Xscl is way too small, and the tickmarks are running together in lines. Similarly, Yscl is the number of y units between tick marks. Abunch of vertical lines means your Yscl is too small. Press [WINDOW] and ix either of these problems.To turn the grid on or off: Locate yellow FORMAT above the [ZOOM] key. Press [2nd ZOOM makes FORMAT]. Cursor to the desired GridOn or GridOff setting, and press [ENTER] to lock it in. Then press [GRAPH] to return to your graph.Exploring Your Graph Domain and Asymptotes First off, just look at the shape of the graph. A vertical asymptote should stick out like a sore thumb, such as x = 3 with this function. (Con irm vertical asymptotes by checking the function de inition. Putting x = 3 in the function de inition makes the denominator equal zero, which tells you that you have an asymptote.) The domain certainly excludes any x values where there are vertical asymptotes. But additional values may also be excluded, even ifthey’re not so obvious. For instance, the graph of f(x) = (x³+1)/(x+1) looks like a simple parabola, butthe domain does not include x = −1. Horizontal asymptotes are usually obvious. But sometimes an apparent asymptote really isn’tone, just looks like it because your ield of view is too small or too large. Always do some algebra workto con irm the asymptotes. This function seems to have y = 1 as a horizontal asymptote as x gets verysmall or very large, and in fact from the function de inition you can see that that’s true. Function ValuesWhile displaying your graph, press [TRACE] and then the x value you’re interested in. The TI‐83/84 willmove the cursor to that point on the graph, and will display the corresponding y value at the bottom. The x value must be within the current viewing window. If you get the message ERR:INVALID, press[1] for Quit. Then adjust your viewing window and try again. 5 of 7
• InterceptsYou can trace along the graph to ind any intercept. The intercepts of a graph are where it crosses ortouches an axis: x intercept where graph crosses or touches x axis because y = 0 y intercept where graph crosses or touches y axis because x = 0Most often it’s the x intercepts you’re interested in, because the x intercepts of the graph y = f(x) are thesolutions to the equation f(x) = 0, also known as the zeroes of the function.To ind x intercepts: You could naïvely press [TRACE] and cursor left and right, zooming in to make acloser approximation. But it’s much easier to make the TI‐83/84 ind the intercept for you.Locate an x intercept by eye. For Locate yellow CALC above the [TRACE] key. Press [2nd TRACEinstance, this graph seems to have an x makes CALC] [2]. (You select 2:zero because the x interceptsintercept somewhere between x = −3 are zeroes of the function.)and x = −1.Enter the left and right bounds. [(-)] 3 [ENTER] [(-)] 1 [ENTER] There’s no need to make a guess; just press [ENTER] again.Two cautions with x intercepts: Since the TI‐83/84 does approximations, you must always check the TI‐83/84 answer in the function de inition to make sure that y comes out exactly 0. When you ind x intercepts, make sure to ind all of them. This particular function has only one in its entire domain, but with other functions you may have to look for additional x intercepts outside the viewing area.Finding the y intercept is even easier: press [TRACE] 0 and read off the yintercept. This y intercept looks like it’s about −2/3, and by plugging x = 0 inthe function de inition you see that the intercept is exactly −2/3.Multiple FunctionsYou can plot multiple functions on the same screen. Simply press [Y=] and enter the second function 6 of 7
• next to Y2=. Press [GRAPH] to see the two graphs together. To select which function to trace along, press [▲] or [▼]. The upper left corner shows which functionyou’re tracing. IntersectionWhen you graph multiple functions on the same set of axes, you can have the TI‐83/84 tell you wherethe graphs intersect. This is equivalent to solving a system of equations graphically. The naïve approach is to trace along one graph until it crosses the other, but again you can dobetter. We’ll illustrate by inding the intersections of y =(6/5)x−8 with the function we’ve alreadygraphed.Graph both functions on the same set of Press [2nd TRACE makes CALC] [5].axes. Zoom out if necessary to ind allsolutions. You’ll be prompted First curve? If necessary, press [▲] or [▼] to select one of the curves you’re interested in. Press [ENTER]. You’ll be prompted Second curve? If necessary, press [▲] or [▼] to select the other curve you’re interested in. Press [ENTER].Eyeball an approximate solution. For When prompted Guess?, enterinstance, in this graph there seems to be your guess. In this case, sincea solution around x = 2. your guess is 2 you should press 2 [ENTER].Repeat for any other solutions.As always, you should con irm apparent solutions by substituting in both equations. The TI‐83/84 usesa method of successive approximations, which may create an ugly decimal when in fact there’s an exactsolution as a fraction or radical.This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/ 7 of 7
• TC3 → Stan Brown → TI-83/84/89 → Piecewise Functions revised 4 Oct 2008 Graphing Piecewise Functions on TI-83/84 Copyright © 2003–2012 by Stan Brown, Oak Road SystemsSummary: You can graph piecewise functions on your TI‐83/84 by using the TEST menu. To show the method, we’ll graph the function which is read “f of x equals x²+11 for x<0, 11−4x for x between 0 and 2 inclusive, and x²−3x+5 for x>2.” This particular function, as you’ll see, doesnt have any gaps in it, but exactly the same technique works for piecewise functions that do have gaps.See also: Graphing Functions on TI‐83/84Set-up: Dot ModeThe TI‐83/84 likes to connect dots with [MODE] [▼ 4 times] [►] [ENTER]continuous lines or curves where it can.But a piecewise function could havegaps legitimately, and therefore youwant to select dot mode.The TI‐83 and TI‐84 MODE screens areslightly different, but the settings arethe same.(You may need to switch between dot mode and connected mode, depending on the functions you’regraphing, because a function with a steeply sloping graph will be hard to see in dot mode.)Enter the FunctionThe general form you’re going for is ( irst piece)( irst condition)+(second piece)(second condition)+... 1 of 3
• This works because in the TI programming language a true condition is equivalent to a 1 and a falsecondition to a zero. Therefore each branch of the function is turned on (multiplied by 1) in the properregion and turned off (multiplied by 0) everywhere else. You can have as many (piece)(condition) pairs as it takes to de ine the function, and you alwaysneed the parentheses around each piece and around each condition. If you have a compound conditionlike 0 ≤ x ≤ 2, you can use [2nd MATH makes TEST] [►] [1] to create an and condition, or code the twoconditions inparentheses and multiply them.For our sample function, you want to get this onto the Y= screen: Y1=(x²+11)(x<0)+(11−4x)(0≤x and x≤2)+(x²−3x+5)(x>2)or Y1=(x²+11)(x<0)+(11−4x)(0≤x)(x≤2)+(x²−3x+5)(x>2)You already know how to do all of that except the inequality signs in the tests, and as you’ll see, that’spretty easy.Clear any previous plots. (Review this [Y=] and deactivate anything that’s highlighted.on the general graphing page if youneed to.)Enter the irst branch of the function On the Y= screen, cursor to one of the Y= lines. Press [CLEAR]de inition, (x²+11). if necessary, and enter the irst piece in parentheses: [(] [x,T,θ,n] [x²] [+] 11 [)]Enter the test, (x<0). Press [(] [x,T,θ,n] [2nd MATH makes TEST] [5] 0 [)]Enter the second branch of the function [+] [(] 11 [−] 4 [x,T,θ,n] [)]de inition, (11−4x).Enter the second test, (0 ≤ x ≤ 2). You [(] 0 [2nd MATH makes TEST] [6]can code this either as the product of [x,T,θ,n] [)] [(] [x,T,θ,n]two tests, (0≤x)(x≤2), or with an and [2nd MATH makes TEST] [6] 2 [)]condition, (0≤x and x≤2). The irst waysaves a couple of keystrokes, so that’swhat I’ll do.Enter a plus sign and the last branch of [+] [(] [x,T,θ,n] [x²] [−] 3 [x,T,θ,n] [+] 5 [)]the function, (x²−3x+5).Enter the last test, (x>2). [(] [x,T,θ,n] [2nd MATH makes TEST] [3] 2 [)] 2 of 3
• Display the GraphIt’s often helpful to start with [ZOOM] [6], standard zoom, and then adjust the window. This particularfunction, I think, is a little easier to visualize with the window parameters shown.You can zoom, trace, and ind values and intercepts just as you would do for any other function. See the general graphing page for common problems. One particular problem with piecewise functions is that the TI‐83/84 may try to connect thepieces. Make sure you are in dot mode, not connected mode: look on the Y= screen for three dots to theleft of your equation.What’s New 4 Oct 2008: change the function from two to three parts, to illustrate all three types of conditions; explain how it works; explain compound conditions; add TI‐84 mode screen 26 Oct 2007: clarify TI‐83/84 keystrokes involving the [2nd] key (various formatting improvements, suppressed) 31 Mar 2003: new documentThis page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/ 3 of 3
• TC3 → Stan Brown → TI-83/84/89 → Inverse Functions revised 15 Feb 2009 (What’s New?) Inverse Functions on TI-83/84 Copyright © 2001–2012 by Stan Brown, Oak Road SystemsGraph an Inverse FunctionSummary: After you graph a function on your TI‐83/84, you can make a picture of its inverse by using the DrawInv command on the DRAW menu.For this illustration, let’s use f(x) = √(x−2), shown at right. Though youcan easily ind the inverse of this particular function algebraically, thetechniques on this page will work for any function. I’ve compensated for the rectangular viewing window by settingwindow margins to 0 to 10 in the x direction and 0 to 6.5 in the ydirection. (If you don’t know how to graph a function, please review thatprocedure.) The graph of an inverse function is a mirror image of the original through the line y=x, so I’ve havealso plotted that line.To draw the inverse of that function:Paste the DrawInv command to your [2nd PRGM makes DRAW]home screen. Either cursor down to the 8 and press [ENTER], or simply press [8].Tell the TI‐83/84 to ind the original Press [VARS] [►] [1] [1].function in Y1. (If your function was in a different numbered y variable, pick that one instead of Y1.)At this point your screen shows thiscommand: DrawInv Y1Now execute the command. Press [ENTER].The result is shown at right.You know from your algebra work that the inverse of f(x) = √(x−2) 1 of 4
• is f‐1(x) = x²+2, x ≥ 0and the graph con irms that.Each point on the graph of f(x) has a corresponding point on the graph of f‐1(x). For example, f(2) = 0,so (2,0) is on the original graph, (0,2) is on the graph of f‐1(x), and f‐1(0) = 2.Unfortunately, all you can do with the inverse is look at it. You can’t trace or do other things. But eventhat helps you check your work. For instance, you see that the inverse of the sample function appearsonly in the positive x region. The inverse you calculate algebraically, x²+2, has a domain in both thepositive and negative reals, but from drawing the inverse on the TI‐83/84 you can see that you need torestrict the inverse function’s domain to match the restricted range of the original function.There’s another way you can check your work. Find the inverse function irst, algebraically, and graph itas Y3 when you graph the original as Y1. If you do that, DrawInv Y1 will exactly overlay the graph ofyour algebraic inverse. Caution: Because the screen resolution is low, two different functions sometimes look the same.This method isn’t an absolute guarantee that your work is correct, but it’s better than no check at all.Find the Value of an Inverse FunctionNow suppose you have to ind f‐1(1.5)? Of course you can look at it on the graph and estimate, but yourcalculator can do a better job of the estimation for you. There are two methods, one on the graph andone on the home screen. Method 1: intersect on GraphRemember that f‐1(1.5) is some value, call it a, such that (1.5,a) is on thegraph of f‐1(x), and therefore (a,1.5) is on the graph of f(x). In otherwords, f‐1(1.5) is the x value on the original graph of f(x) where the yvalue is 1.5. Using this idea, to ind f‐1(1.5) you can plot y = 1.5 and have yourcalculator ind the point where it intersects the graph of f(x). You don’tneed the graph of f‐1(x) for this at all. The graphs are shown at right, and here’s the procedure.Select the intersect command. [2nd TRACE makes CALC] [5]The calculator asks “First curve?” Simply press [ENTER] to select the irst curve. 2 of 4
• The calculator then asks “Second The cursor may have moved automatically to the othercurve?” curve. If not, press [▲] or [▼] until it does. Press [ENTER].Finally, the calculator asks for your Usually you can just press [ENTER]. But if the function is veryguess. complicated, you can use [◄] or [►] to move the cursor close to the intersection point and then press [ENTER], or type in a number and press [ENTER].The result is shown at right: the answer is 4.25. Why is the answer x and not y? Because you’re trying to ind f‐1(1.5),the value of the inverse function of 1.5. But as mentioned above, f-1(1.5)is the number a such that f(a) = 1.5. In other words, becausef(4.25) = 1.5, f‐1(1.5) = 4.25.Caution: Your calculator gives numerical solutions only. To determinewhether 4.25 is the exact answer or just a good approximation, you haveto check it in the original function. Method 2: solve on Home ScreenYou can accomplish the same thing on the home screen by using the solve function.Select the solve function from the Press [2nd 0 makes CATALOG] [ALPHA 4 makes T], scroll up tocatalog because it’s not in a menu. solve(, and press [ENTER].(There’s a Solver command in the Mathmenu, but setting it up is a little morework.)The irst argument is an expression that Press [VARS] [►] [1] [1] [−] [1] [.] [5]you want to equate to zero. You actuallywant to equate Y1 to 1.5, which is thesame as equating Y1−1.5 to 0.The second argument is the variable, x. Press [,] [x,T,θ,n]. 3 of 4
• The last argument is your initial guess. Enter the initial guess and a close parenthesis [)].Unless the function is prettycomplicated, it doesn’t matter what youenter here as long as it’s in the domainof the function. For example, 0 would bea bad choice for f(x) = √(x−2) becausef(0) is not a real number. Let’s use 6 asthe initial guess.The screen is shown at right. The answer of 4.25 agrees with thegraphical method. Caution: Again, remember that this is a numerical solution andmay not be exact.What’s New 15 Feb 2009: Add section, suggested by Max Harwood, on inding value of inverse function; edit the graphing section extensively for clarity; drop “drawing” from document title (intervening changes suppressed) 7 Jun 2003: irst publication, as “Drawing Inverse Functions on the TI‐83”This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/ 4 of 4