Graphing an inequality on the TI-89

The graphical solution to Example 14 Chapter 1 Section 2 of the text is not quite as simple to construct as one might hope. That example uses the problem

Figure 1	Figure 1 shows the result of pressing the keys on a TI-89 calculator. The actual output depends on the previous state of the calculator. In this example, it would appear that no functions have been defined and that the window settings are not at all standard.
Figure 2	Press the keys to see the "y(x)=" list. On this calculator that list is empty. The calculator is ready to receive the first function definition.
Figure 3	To produce Figure 3 we have pressed the and keys to create the first function, which appears on the data entry line at the bottom of the screen.
Figure 4	Then we press the key to move to complete the entry of the first function. This places the first function in the "y1=" line of the display. In addition, the calculator is now ready for the second function. That function is created by the and keys. Again, we press the key to accept the second function. It, too, is displayed at the top, in the "y2=" line.
Figure 5	Having entered the functions in Figures 3 and 4, we press to select the GRAPH command. The result is shown in Figure 5. We can see the graphs of the two straight lines in Figure 5, but this graph does not appear as does the graph in the text. Part of that difference is due to the greater screen resolution available on the TI-89. However, the fact that the point of intersection for the two lines is not visible and the fact that the lines seem to be at different angles than the lines in the textbook graph are related to the WINDOW settings, not to the finer resolution of the TI-89. We will look at the WINDOW settings to see the current values and to change those values if need be.
Figure 6	To open the WINDOW menu we press the key. The result is shown in Figure 6. Clearly these are not the values that we want to be using. Rather than type all new values, let us use the ZOOM menu to change the WINDOW settings. To do this, press the key to select the ZOOM option shown at the top of the screen. This will open a small menu box, as is shown in Figure 7.
Figure 7	We will opt for the ZoomStd option by pressing . This will set the xMin value to – 10, xMax to 10, xScl to 1, yMin to – 10, yMax to 10, and yScl to 1. It also moves us back to graph mode and a new graph, see Figure 8, will be drawn.
Figure 8	In Figure 8 we have the new graph that uses the new WINDOW settings. It still does not seem to correspond to the figure in the book. First, the x-axis in Figure 8 needs to be raised. We can do this by making the yMin value more negative. Second, the "tick" marks on the y-axis are too close together. We can alter this by increasing the value assigned to yScl.
Figure 9	We move to Figure 9 by presing the key to open the WINDOW settings. A review of Figure 9 shows that we have indeed established the ZoomStd settings described above.
Figure 10	Figure 10 shows the WINDOW settings screen after we have used the cursor keys to move down to the yMin line, and then we entered a new value for yMin, namely, – 20. Then we moved to the yScl line and changed that value to be 2.
Figure 11	Now, to get to Figure 11 we press the key to do another graph. This graph looks similar to the graph in the textbook. However, the graph in the text tells us the x and y coordinate of the point of intersection of the two lines. We need to find the command to get the calcualtor to find that point. A review of the menu at the top of Figure 11 does show that we have the "Math" option available. We press the key to open that option window.
Figure 12	The Math window is displayed in Figure 12. The command that we want to use is item 5: Intersection. We press the key to select that option.
Figure 13	As a result of selecting the Intersection option, Figure 13 shows us that the calculator is proposing that our line for y(x)=5x+2 be our first curve. We can examine the graph in Figure 13 and we will see that, at the point x=.12658228, y=2.6329114, the calcualtor has displayed a special symbol. That is how the calculator identifies the particular line that it is proposing to use. We can press the key to accept that proposal. This will move the graph to Figure 14.
Figure 14	Now we need to select the second curve. The calculator is proposing the other line, y(x)=x–6, for the second curve. The calculator is making this proposal by displaying its flashing sign on a point, x=0.12658228, y=– 5.873418, on that line. Again, we will accept the proposal by pressing the key.
Figure 15	In Figure 15 the calculator has been given the two curves, and now it wants a a lower bound, the low end of the interval in which it should look for the intersection. The calcualtor offers the point (0.12658228,– 5.873418) as that lower bound. We do not want to use that value. Therefore, we will continue to press the key to move the flashing target to the left of the point of intersection. The result is shown in Figure 16.
Figure 16	In Figure 16 we have moved the flashing target to the left of the point of intersection. This new point, (-3.6708886,– 9.670886), can be our lower bound. Therefore, we press the key to accept that point.
Figure 17	The calculator responds, as shown in Figure 17, by asking for an Upper Bound. (In addition, the calculator has placed a small right-pointing arrow just below the top option line, to indicate were we have set the lower bound.) Again the proposed point is not acceptable. We use the key to move the flashing target to the right until we have moved beyond the point of intersection.
Figure 18	Having moved the flashing target to the right, we are now in a position to accept (0.63291139,– 5,367089) as the Upper Bound. We do this by pressing the key.
Figure 19	As a result of all of our efforts, the calculator display shifts to Figure 18. In that Figure, the calculator has identified the point of intersection of the two lines, namely at the point (– 2,– 8). Even given the better resolution on the TI-89, this graph is remarkably similar to the graph that is given in the textbook. The major difference is that the textbook version contains the equations of the two lines. A closer examination of the graph in the textbook shows that these equations were pasted into the book; after all, the equations in the book appear in a completely different font. We too can doctor a picture. We have done that in Figure 20.
Figure 20	You can not produce Figure 18 directly from the calculator. The equations have been added to this image. In addition, we have added a red line from the point of intersection straight up to the x-axis. In additon, we have replaced the x-axis from directly above the point of intersection all the way to the left by a light blue line. This represents the x-values where the line y(x)=5x+2 is below the line y(x)=x–6. In effect, this is the set of points where 5x+2 < x-6 which was the original problem.

The material above produced a graph similar to the one given in the textbook. The detailed TI-86 notes for this section address the steps needed to produce a graph directly from the given inequality, namely,

Figure 21	Figure 21 returns to the y= screen via the keys. In this case we have cleared all of the old functions and have added the line that you see in Figure 21, namely, y1=tester("5x+2<x–6",x) This is the method used to call the function tester() and to send it the relational expression 5x+2<x–6. The additional ,x is required.
Figure 22	In Figure 22 we graph the new function by pressing the keys. We can see that the value of the function is 1 up to x having the value of negative two (– 2. At that point the graph drops to 0, which is right on top of the x-axis. This is a problem similar to that seen for the TI-86. We can return to the y= screen via the key sequence and there we can modify the function to use the same trick that we did for the TI-86.
Figure 23	Back at the y= screen, we press the key to bring a copy of the y1 function into the edit area at the bottom of the screen. Then we modify the function so that it appears as y1=tester("5x+2<x–6",x)*5–2 Then, we press the key to accept the new definition for y1. This leaves us with the screen appearing as in Figure 23. Note that there is not sufficient roomn to fully display the new function on the y1= line.
Figure 24	In Figure 24 we graph the new function by pressing the keys. Now we can see more clearly the region where the function is true (the elevated portion of the graphed values) and where it is false (the depressed portion of the graphed values). There does seem to be an oddity here. The graph does not drop down from true to false immediately. A close examination of the graph indicates that the drop happens over a few pixels. This should not be the case. We know that the value of the function is either 3 for TRUE or – 2 for FALSE. What is going on here?
Figure 25	In Figure 25 we have returned to the WINDOW settings screen via the keys. When we last saw this screen, back in Figure 10, we did not pay any attention to the final setting, the value assigned to xres. That values was 2. Now, in Figure 25, we have moved down to that value and have changed it to be 1. The xres setting determines how often the calculator will compute values from the domain of the function. When xres was set at 2, the TI-89 would compute function values for every second point on the graph, and the calculator would interpolate values for the missing points. Doing this means that the calculator only has to evaluate the function for half of the points on the graph. This speeds up the process of drawing the graphs. Now that we have changed the xres setting to be 1, on any new graph the calculator will compute the value of the function at each point on the x-axis.
Figure 26	We return to the graph via the keys. This time the drop from TRUE to FALSE is much more immediate.
Figure 27	The following 8 screens demonstrate the steps needed to obtain a listing of the tester() function. We start by pressing the keys. This opens the window shown in Figure 27. Of these applications, we wish to use the Program Editor.
Figure 28	In Figure 28 we have selected the Program Editor by pressing the key. This opens another window, one that asks if we want to open the Current program, some other program, or create a new program? We have moved the highlight to the second option, to open some other program. Then, to perform that option we press the key.
Figure 29	Figure 29 shows the screen that will allow us to choose the program that we wish to open. In this case, we will want to open the function tester(). A function is a special kind of program. The setting for Figure 29 is to open a program. We need to change this to open a function. Therefore, we press the key to open the selection window, as shown in Figure 30.
Figure 30	The selection window in Figure 30 gives us the choice of a program or a function. Naturally, we select option 2, function, by pressing the key.
Figure 31	Notice, in Figure 31, that the screen now indicates that we are looking to open a function. That function happens to be in the main folder, so the next setting is correct. But the bottom setting suggests that the name of the function that we want to open in main is piece. We need to change this setting.
Figure 32	In Figure 32, we have pressed the key to look at the list of functions that we can open. Then we have moved the highlight down to tester. We can select that function by pressing the key. This will move us to Figure 33.
Figure 33	In Figure 33 we finally have our options set appropriately. We press the key to perform the task of openning the function tester in the main folder.
Figure 34	Figure 34 lists the function tester. This is the text that you would have to type into the calculator if you wanted to create your own version of tester without downloading it from another machine or from a PC.