MATH 169: Chapter 3: Section 2: Demo

Chapter 3, section 2 starts to use the calculator to do graphs of functions. Although these graphs ae helpful in understanding the functions being used, the book does not take the time and space to explain how one can generate the graphs. In addition, the text presents a number of real world examples that are discontinuous and yet the graph presented is of a continuous nature. It is possible to generate a discontinuous graph on the TI-83, but it is not a pretty solution. Rather, it makes more sense to "plot" a table of values rather than to "graph" a function in these cases. The first example below does this. We start with a number of images that set up the calculator for the task at hand (Figures 1 through 11). Then, in Figures 12 through 35, this page takes the problem from the question 1 on page 145 and it creates a "plot" of the discontinuous values for the function for the values 1 through 9 credits. After that sequence of steps, this page graphs the continuous function y1=7500 on top of the plot of the discontinuous points (in Figures 36 through 38). After Figure 38, we take a small diversion to actually look at the steps needed to generate a discontinuous function for our problem. This is done in Figures 38a through 38h. This is an amazingly ugly process. It may be reasonable to actually skip Figures 38a through 38, assuming one is not interested in this more elaborate use of the the TI-83 calculator.

Figure 39 is assumed to follow Figure 38. In Figures 39 through 49 we go on to develop the graph at the bottom of page 147. Although the problem should be discontinuous, the graph is the calculator version of a continous set of values for y1=8.5x–800. The real trick here is that the graph is done on a split screen. It only takes a few steps to shift the calculator to split screen mode, but it takes a bit of planning to be able to get the screen and the exact table given at the bottom of page 147.

Figure 1 The first 11 images on this page walk us through a process to set up the calculator into a default mode. We do this so that when we do the real work of the page (starting on Figure 12) you can follow along on your calculator.
We start by turning off any "plots" that we may have on the calculator. We do this by moving to the STAT PLOTS screen, shown in Figure 1, by pressing the keys.
Figure 2 We want to select the fourth option from Figure 1, therefore we press . This will paste the command PlotsOff onto the screen, shown in Figure 2.

Figure 3 We press to perform the PlotsOff. The calculator responds with Done. That command turns off all "plots", which was our intent.

Figure 4 Next, we would like to clear all of the lists that are defined on the calculator. There is a command to do this but we will have to find it in the catalog. We open the catalog by pressing the keys. This starts at the beginning of the catalog.

Figure 5 The command that we want is called ClrAllLists. We could use the down arrow to move to that command. However, we will take the short-cut of pressing to move to the start of the "C's" in the calatog. Strangely enough, the first "C" command is . This is a command that starts with the Greek letter "chi", which gives us an indication of why we find it here. Then we press to move the indicator arrow down to our desire command, ClrAllLists. This is the condition shown in Figure 5.

Figure 6 We press to select the command fromthe catalog listing in Figure 5. This will pate the command to the screen. Then we press to actually perform the command, as shown in Figure 6. At this point we have cleared any list that had been defined in the calculator.

Figure 7 Before we start the real work that we want to do, we need to be sure that we do not have any functions already defined. On the calculator used to generate these Figures, we press to open the Y= screen. Figure 7 demonstrates that one function is already defined on this calculator.

Figure 8 We press to clear that function.

Figure 9 And, then we press to quit the Y= screen of Figure 8 and return to the main screen, Figure 9.

Figure 10 Our final preparation will be to set up the list editor. We do this by moving to the STAT screen via the key. On that screen, shown in Figure 10, tour desired command, SetUpEditor, is option number 5.

Figure 11 We select that option by pressing , which will paste the command onto our main screen. Then we press to actually perform the operation.
SetUpEditor initializes the Stat Editor so that it uses the built-in lists, L1 through L6.

That completes the steps needed to initialize the calculator. Of course, these steps could have been done in a different order. In addition, some of the steps may not have been needed, given that the calculator may already have been set the way we want it.

The problem that we are examining is to express the tuition at a university as a function of the number of credits taken. The tuition is given as $7500, no matter how many credts a student takes. We are lead to believe that a student may only take a whole number of credits, between 1 and 25. In order to simplify the example, the steps below will further restrict that to a number between 1 and 9, inclusive. The expression of the function is

f(x) = 7500 but, we know that there is a limit on the number of credits that a student may take. This is a great place to recall that our function definition, given above, is really shorthand for f = {(x,y)| y = 7500, x is a whole number between 1 and 25} or in or restricted case, where x is a whole number between 1 and 9. For us that means that f ={ (1,7500), (2,7500), (3,7500), ..., (9,7500)} We will use two lists to hold the x and y coordinates of these ordered pairs. Then we can plot those ordered pairs on our graph.

Figure 12 We will start by moving to the STAT screen via the key. This brings up the screen shown in Figure 12. We want to move to the Stat Editor, which is the first option in Figure 12.

Figure 13 Because it is the highlighted option, ee press to select it. This opens the Stat Edit screen shown in Figure 13. Note that the lists that are displayed in Figure 13 are L1, L2, and L3. Those lists are there because of the preparation we took back in Figures 10 and 11. Furthermore, the lists do not hold anything because of the preparation we took in Figures 4 through 6.

Figure 14 We will add our x values to the L1 list by pressing . This results in Figure 14.

Figure 15 Of course, in our restricted problem, we want the x values to continue through 9. We press to add the rest of the values. The display shifts down to allow us to enter each new value. We have ended the entry of our x values.

Figure 16 To add our y values we need to move to the L2 list. We can do this by pressing the key. The highlight moves to the L2 column, at the top of that column since there are no values defined for L2 at this time.

Figure 17 We could enter the value 7500 into each of the cells of L2. In fact, by using Figure 17 demonstrates such a process.
We could continue this pattern to enter the remaining 7 values for our restricted version. It would have been harder to enter 25 values for the original problem. However, here we will use another method that would be just as easy to enter 25 values or even 125 values.

Figure 18 First we will get out of the STAT editor by pressing . This returns us to the main screen.

Figure 19 We had left the Stat Editor when we had 9 values in L1 but only 2 in L2. We want to make L2 have 9 elements. One way to do this is to just copy L1 to L2. We use the key sequence to set up the command to do this.

Figure 20 Then we press to perform the command. The resulting values in L2 are shown in Figure 20. Of course, we do not want those values. Rather, we want every value in L2 to be 7500. There is a command to do this. But we need to find that command.

Figure 21 We open the LIST screen by pressing .

Figure 22 We want one of the LIST related commands. Therefore, we press to move the highlight to the OPS choice. The calculator responds by displaying the choices shown in Figure 22. We are interested in the Fill( command.

Figure 23 We can press to select that Fill( command and paste it onto the main screen, shown in Figure 23.

Figure 24 In Figure 24 we complete the Fill( command by supplying the value to use, 7500, and the name of the list to fill, L2. We use the keys to do this.

Figure 25 As usual, we press to perform the command that we created in Figure 25.

Figure 26 We can check on our work by returning to the Stat Editor. Press to move to Figure 26.

Figure 27 The key will open the Stat Editor, shown in Figure 27. The desired values are in the desired lists. It would look as if we are ready to get our graph.

Figure 28 The will move us to the graph screen.
This is not what we wanted. What went wrong?
First, our WINDOW settings are clearly out of the required range.

Figure 29 The key opens the WINDOW screen. As shown in Figure 29, these values are clearly too limiting. There is no way that we will see Y values such as 7500 when the Ymax is 10.

Figure 30 We will change the settings by using the keys and . This will produce Figure 30.

Figure 31 Pressing the key takes the calculator back to the graph screen. Our change in the WINDOW settings seems to have taken effect, but our valeus are still not there. Now what is wrong?
We have a graph in Figure 31, but we needed to "plot" the two lists! And, in Figures 1 through 3 we had turned off all of the plots. It must be time to turn one of them on again.

Figure 32 To get back to the STAT PLOT screen, press . The resulting Figure 32 indicates that all the plots are indeed "Off". Let us turn on the first plot.

Figure 33 The first plot was the highlighted plot in Figure 32. Thus, pressing will move the calculator to Figure 33. Note that Figure 33 was captured from a calculator when the blinking cursor, which was on the "On", was hidden. We need to change the setting from "Off" to "On". The other settings in Figure 33 seem to be just fine. The Type of plot will be a scatter plot, rather than a line or histogram plot. Plot1 is set to use L1 for the Xlist and L2 for the Ylist. And the plot character will be a square around the desired value.

Figure 34 We change the setng to "On", the highlighted option by pressing . Figure 34 shows that the setting has been changed.

Figure 35 Again, the returns the calculator to the graph screen. This time, however, that screen contains the plot that we have constructed. This is a good representation of the function for our restricted problem.

Figure 36 We return to the Y= screen via the key. We will define the usual constant function, Y=7500, and then we can return to the graph to see that function.

Figure 37 The sequence gives a constant value to the first function, Y1.

Figure 38 Once more, takes us back to the graph screen. However, this time, we have both the plot that we generated and the constant function that we just defined.

We will take a small diversion here to change the continuous function definition developed in Figure 37 into a discontinuous function. This is extra material, and it may be that the reader would like to skip this fairly advanced use of the TI-83 calculator. Figure 39 will pick up from the point that we have left in Figure 38.

The function

Y = 7500 is defined for every value of X. We want a function that is only defined for the integer values of X. We will use three characteristics of the TI-83 to create such a discontinuous function. First, the calculator refuses to do a division by zero. Second, the TI-83 treats a "true" relational expression as the value 1 and it treats a "false" relational expression as the value 0. And third, the built-in function iPart(X) holds the value of just the integer part of X. This means that X=iPart(X) will be 0 for any value of X that is not an integer, but it will be 1 if X is an integer. Therefore, the function Y = 7500/(X=iPart(X)) will have the value 7500/1 or 7500 whenever X is an integer, but it will represent 7500/0 whenever X is not an integer, and, the TI-83 will refuse to do that division, calling the result "undefined".

It is an interesting and important characteristic of the TI-83 that in doing a graph the TI-83 uses X values that represent the "center" of each pixel moving across the screen. These values, in turn, are determined by the xMin and xMax settings on the WINDOW screen, and by the fact that there 95 pixels across the screen. Therefore, when we leave the WINDOW as it was set in Figure 30, the first X value will be -1 and the last value will be 10. The step from pixel to pixel will be 11/94 (there being 94 steps from the first to the last). In particular, the calculator will never try to evaluate the function for values such as 1, 2, 3, and so on. We need to change the WINDOW settings so that the X values will hit our desired points. In particular, if we set xMin to ^–.4 and xMax to 18.4, then the range of values will be 18.8 units. If we divide 18.8 by 94, we find that each step between pixels will be 0.2, which means that we will "hit" values such as 1, 2, 3, and so on.

Figure 38a We return to the WINDOW screen via the key. Then we set the desired values via the and keys. The result is shown in Figure 38a.

Figure 38b Let us may a quick return to the graph screen, via the key, to see the effect of our changes.

Figure 38c Now we will enter the new function by moving to the Y= screen, via , and then moving to the end of the 7500 by pressing , and then start appending the rest of the function via the keys. To generate the equal sign, we use to open the TEST menu, and then to select the equal sign, which is then pasted into the function. To find the iPart( component, we press to open the MATH menu, press to move to the NUM sub menu, and then select the third item in that menu by pressing the key. Again, the choice is pasted into our function. That leaves us the task of pressing the and keys to complete the function.

Figure 38d We return to the graph screen by pressing . This graph is captured in Figure 38d. Note that the "plot" squares are filled in because we have "graphed" the points inside them. The new points, the ones to the right of the "plot" points, represent more of the discontinuous function graph.

Figure 38e To further support and demonstrate our actions, press to shift the calculator into TRACE mode. Figure 38e shows that we start by tracing the "plot", as shown by the displayed in the upper left corner of the graph.

Figure 38f We can press to change the TRACE to follow the function definition. Note the change in Figure 38f. Also, in Figure 38f, we note that the trace is at the point where X=9 and Y=7500.

Figure 38g If we press the new value of X becomes 9.2. However, when X=9.2 the function is "undefined". Note the display at the bottom of Figure 38g.

Figure 38h If we press we shift the X value to 10. Now the function is defined. The TI-83 displays both the value of Y and it highlights the point on the screen.

We return, at this point, to the state of the calculator after Figure 38.

Our next challene will be to produce the screen image given at teh bottom of page 147. That image has a split screen, one showing both the graph and a supporting table. It does not take many steps to shift the calculator into such a split screen mode. It will take a bit of work to really produce the results at the bottom of page 147.

Figure 39 We start by moving to the MODE screen. We do this by pressing the key. The value that we want to change is the last one on the screen.

Figure 40 We move down to that line by pressing and . Then we press to move the blinking cursor over the G-T option. Finally, we press to change the selection to be G-T, the option requesting a Graph and a Table.

Figure 41 We return to the Graph window by using the key. In Figure 41 we note that we do have the desired split screen, but it is showing the values that we had earlier. We need to change the function definition to that of page 147.

Figure 42 The key returns us to the Y= screen.

Figure 43 We can clear the existing definition via the key. Then we enter the new function Y=8.5 – 800 by pressing

Figure 44 We can check our fforts by going to the graph window via the key. The result is shown in Figure 44. The graph does not show our function. However, we can see why when we look at the table. There we note that for small integer values of X the value of the function is quite negative. Our graph shows X values from -1 to -10. The graph of the Y values is off the screen.

Figure 45 We need to change the WINDOW settings. Press to return to the WINDOW screen.

Figure 46 From Figure 45 we know that we need to change the WINDOW settings. But what values should we use? For one thing, we note that the chart on page 147 has a step of 7 between X values. In the split screen mode, the TI-83 uses 47 pixels across the graph. Thus, there are 46 steps from pixel to pixel. If we want each of those steps to represent a change of 7, then the range of our X values will need to be 7*46 or 322.
The settings for the Y values are not obvious. We will use the values ^–1000 to 1000, even though this is not quite the values in the book. Figure 46 reflects these changes in the WINDOW settings.

Figure 47 We use to return to the graph window. Figure 47 shows the result. Note that the graph looks reasonably correct, but the table is unchanged. In fact, the steps between X values is still 1 in Figure 47.

Figure 48 However, in Figure 48 we move to TRACE mode by pressing the key. Now, in Figure 48, the steps between X values in the table is at our desired 7, however, the wrong values are being displayed.

Figure 49 Of course, we can trace back on the graph by pressing again and again until we have moved the highlight to the point where X=84. Then, if we press two times we will have the screen shown in Figure 49.

Figure 1	The first 11 images on this page walk us through a process to set up the calculator into a default mode. We do this so that when we do the real work of the page (starting on Figure 12) you can follow along on your calculator. We start by turning off any "plots" that we may have on the calculator. We do this by moving to the STAT PLOTS screen, shown in Figure 1, by pressing the keys.
Figure 2	We want to select the fourth option from Figure 1, therefore we press . This will paste the command PlotsOff onto the screen, shown in Figure 2.
Figure 3	We press to perform the PlotsOff. The calculator responds with Done. That command turns off all "plots", which was our intent.
Figure 4	Next, we would like to clear all of the lists that are defined on the calculator. There is a command to do this but we will have to find it in the catalog. We open the catalog by pressing the keys. This starts at the beginning of the catalog.
Figure 5	The command that we want is called ClrAllLists. We could use the down arrow to move to that command. However, we will take the short-cut of pressing to move to the start of the "C's" in the calatog. Strangely enough, the first "C" command is . This is a command that starts with the Greek letter "chi", which gives us an indication of why we find it here. Then we press to move the indicator arrow down to our desire command, ClrAllLists. This is the condition shown in Figure 5.
Figure 6	We press to select the command fromthe catalog listing in Figure 5. This will pate the command to the screen. Then we press to actually perform the command, as shown in Figure 6. At this point we have cleared any list that had been defined in the calculator.
Figure 7	Before we start the real work that we want to do, we need to be sure that we do not have any functions already defined. On the calculator used to generate these Figures, we press to open the Y= screen. Figure 7 demonstrates that one function is already defined on this calculator.
Figure 8	We press to clear that function.
Figure 9	And, then we press to quit the Y= screen of Figure 8 and return to the main screen, Figure 9.
Figure 10	Our final preparation will be to set up the list editor. We do this by moving to the STAT screen via the key. On that screen, shown in Figure 10, tour desired command, SetUpEditor, is option number 5.
Figure 11	We select that option by pressing , which will paste the command onto our main screen. Then we press to actually perform the operation. SetUpEditor initializes the Stat Editor so that it uses the built-in lists, L1 through L6.

Figure 12	We will start by moving to the STAT screen via the key. This brings up the screen shown in Figure 12. We want to move to the Stat Editor, which is the first option in Figure 12.
Figure 13	Because it is the highlighted option, ee press to select it. This opens the Stat Edit screen shown in Figure 13. Note that the lists that are displayed in Figure 13 are L1, L2, and L3. Those lists are there because of the preparation we took back in Figures 10 and 11. Furthermore, the lists do not hold anything because of the preparation we took in Figures 4 through 6.
Figure 14	We will add our x values to the L1 list by pressing . This results in Figure 14.
Figure 15	Of course, in our restricted problem, we want the x values to continue through 9. We press to add the rest of the values. The display shifts down to allow us to enter each new value. We have ended the entry of our x values.
Figure 16	To add our y values we need to move to the L2 list. We can do this by pressing the key. The highlight moves to the L2 column, at the top of that column since there are no values defined for L2 at this time.
Figure 17	We could enter the value 7500 into each of the cells of L2. In fact, by using Figure 17 demonstrates such a process. We could continue this pattern to enter the remaining 7 values for our restricted version. It would have been harder to enter 25 values for the original problem. However, here we will use another method that would be just as easy to enter 25 values or even 125 values.
Figure 18	First we will get out of the STAT editor by pressing . This returns us to the main screen.
Figure 19	We had left the Stat Editor when we had 9 values in L1 but only 2 in L2. We want to make L2 have 9 elements. One way to do this is to just copy L1 to L2. We use the key sequence to set up the command to do this.
Figure 20	Then we press to perform the command. The resulting values in L2 are shown in Figure 20. Of course, we do not want those values. Rather, we want every value in L2 to be 7500. There is a command to do this. But we need to find that command.
Figure 21	We open the LIST screen by pressing .
Figure 22	We want one of the LIST related commands. Therefore, we press to move the highlight to the OPS choice. The calculator responds by displaying the choices shown in Figure 22. We are interested in the Fill( command.
Figure 23	We can press to select that Fill( command and paste it onto the main screen, shown in Figure 23.
Figure 24	In Figure 24 we complete the Fill( command by supplying the value to use, 7500, and the name of the list to fill, L2. We use the keys to do this.
Figure 25	As usual, we press to perform the command that we created in Figure 25.
Figure 26	We can check on our work by returning to the Stat Editor. Press to move to Figure 26.
Figure 27	The key will open the Stat Editor, shown in Figure 27. The desired values are in the desired lists. It would look as if we are ready to get our graph.
Figure 28	The will move us to the graph screen. This is not what we wanted. What went wrong? First, our WINDOW settings are clearly out of the required range.
Figure 29	The key opens the WINDOW screen. As shown in Figure 29, these values are clearly too limiting. There is no way that we will see Y values such as 7500 when the Ymax is 10.
Figure 30	We will change the settings by using the keys and . This will produce Figure 30.
Figure 31	Pressing the key takes the calculator back to the graph screen. Our change in the WINDOW settings seems to have taken effect, but our valeus are still not there. Now what is wrong? We have a graph in Figure 31, but we needed to "plot" the two lists! And, in Figures 1 through 3 we had turned off all of the plots. It must be time to turn one of them on again.
Figure 32	To get back to the STAT PLOT screen, press . The resulting Figure 32 indicates that all the plots are indeed "Off". Let us turn on the first plot.
Figure 33	The first plot was the highlighted plot in Figure 32. Thus, pressing will move the calculator to Figure 33. Note that Figure 33 was captured from a calculator when the blinking cursor, which was on the "On", was hidden. We need to change the setting from "Off" to "On". The other settings in Figure 33 seem to be just fine. The Type of plot will be a scatter plot, rather than a line or histogram plot. Plot1 is set to use L1 for the Xlist and L2 for the Ylist. And the plot character will be a square around the desired value.
Figure 34	We change the setng to "On", the highlighted option by pressing . Figure 34 shows that the setting has been changed.
Figure 35	Again, the returns the calculator to the graph screen. This time, however, that screen contains the plot that we have constructed. This is a good representation of the function for our restricted problem.
Figure 36	We return to the Y= screen via the key. We will define the usual constant function, Y=7500, and then we can return to the graph to see that function.
Figure 37	The sequence gives a constant value to the first function, Y1.
Figure 38	Once more, takes us back to the graph screen. However, this time, we have both the plot that we generated and the constant function that we just defined.

Figure 38a	We return to the WINDOW screen via the key. Then we set the desired values via the and keys. The result is shown in Figure 38a.
Figure 38b	Let us may a quick return to the graph screen, via the key, to see the effect of our changes.
Figure 38c	Now we will enter the new function by moving to the Y= screen, via , and then moving to the end of the 7500 by pressing , and then start appending the rest of the function via the keys. To generate the equal sign, we use to open the TEST menu, and then to select the equal sign, which is then pasted into the function. To find the iPart( component, we press to open the MATH menu, press to move to the NUM sub menu, and then select the third item in that menu by pressing the key. Again, the choice is pasted into our function. That leaves us the task of pressing the and keys to complete the function.
Figure 38d	We return to the graph screen by pressing . This graph is captured in Figure 38d. Note that the "plot" squares are filled in because we have "graphed" the points inside them. The new points, the ones to the right of the "plot" points, represent more of the discontinuous function graph.
Figure 38e	To further support and demonstrate our actions, press to shift the calculator into TRACE mode. Figure 38e shows that we start by tracing the "plot", as shown by the displayed in the upper left corner of the graph.
Figure 38f	We can press to change the TRACE to follow the function definition. Note the change in Figure 38f. Also, in Figure 38f, we note that the trace is at the point where X=9 and Y=7500.
Figure 38g	If we press the new value of X becomes 9.2. However, when X=9.2 the function is "undefined". Note the display at the bottom of Figure 38g.
Figure 38h	If we press we shift the X value to 10. Now the function is defined. The TI-83 displays both the value of Y and it highlights the point on the screen.

Figure 39	We start by moving to the MODE screen. We do this by pressing the key. The value that we want to change is the last one on the screen.
Figure 40	We move down to that line by pressing and . Then we press to move the blinking cursor over the G-T option. Finally, we press to change the selection to be G-T, the option requesting a Graph and a Table.
Figure 41	We return to the Graph window by using the key. In Figure 41 we note that we do have the desired split screen, but it is showing the values that we had earlier. We need to change the function definition to that of page 147.
Figure 42	The key returns us to the Y= screen.
Figure 43	We can clear the existing definition via the key. Then we enter the new function Y=8.5 – 800 by pressing
Figure 44	We can check our fforts by going to the graph window via the key. The result is shown in Figure 44. The graph does not show our function. However, we can see why when we look at the table. There we note that for small integer values of X the value of the function is quite negative. Our graph shows X values from -1 to -10. The graph of the Y values is off the screen.
Figure 45	We need to change the WINDOW settings. Press to return to the WINDOW screen.
Figure 46	From Figure 45 we know that we need to change the WINDOW settings. But what values should we use? For one thing, we note that the chart on page 147 has a step of 7 between X values. In the split screen mode, the TI-83 uses 47 pixels across the graph. Thus, there are 46 steps from pixel to pixel. If we want each of those steps to represent a change of 7, then the range of our X values will need to be 746 or 322. The settings for the Y values are not obvious. We will use the values ^–1000 to 1000, even though this is not quite the values in the book. Figure 46 reflects these changes in the WINDOW* settings.
Figure 47	We use to return to the graph window. Figure 47 shows the result. Note that the graph looks reasonably correct, but the table is unchanged. In fact, the steps between X values is still 1 in Figure 47.
Figure 48	However, in Figure 48 we move to TRACE mode by pressing the key. Now, in Figure 48, the steps between X values in the table is at our desired 7, however, the wrong values are being displayed.
Figure 49	Of course, we can trace back on the graph by pressing again and again until we have moved the highlight to the point where X=84. Then, if we press two times we will have the screen shown in Figure 49.