Chapter 1, Section 2, Example 15 on TI-89

The following screen images trace the steps needed to generate a graph on a TI-89 for the solution to the problem given as

3(x + 2) - 5x 4x This solution uses the relational operation

. However, because the TI-89 does not respresent the values TRUE and FALSE by the numbers 1 and 0, the screens here use the tester() function to convert the inequality into the value 1 for TRUE and the value 0 for FALSE.

Figure 1 We have opened the y= menu in Figure 1 by pressing the keys. The calculator used to generate these screens did not have any previously defined graphs on it. Had a graph been defined earlier on the calculator, we would need to CLEAR any old definitions in order to arrive at Figure 1.

Figure 2 We need to from the top portion of the screen in Figure 1 to the function entry and edit line at the bottom. To do this we can press the key. Now we are ready to enter the inequality.

Figure 3 The actual command that we want to enter is tester("3(x+2)–5x4x",x) We can generate the tester by finding it in the VAR-LINK menu, or we could just presss the keys to lock the calculator in alphabetic mode, and then press , followed by to leave alphabetic mode. Then we can continue creating the line shown in Figure 3 via the keys . Now we need to genreate the "less than or equal to" sign. It is not on the keyboard, but we can get it by pressing the keys. And, we complete Figure 3 with the key. The desired command is not done. More text must be entered, and we will do that in Figure 4.

Figure 4 To complete the command started in Figure 3 we press and . The result should be the line shown at the bottom of Figure 4.

Figure 5 Move from Figure 4 to Figure 5 by pressing the key. The calculator is now ready for a second function. We do not have a second function to graph. Instead, we wish to graph the first function on a standard graph. We will move to the ZOOM window to select such a setting. Note the in the menu at the top of the screen.

Figure 6 We open the ZOOM window by pressing the key. The ZOOM window is shown in Figure 6. We want to select the ZoomStd item, item 6, from the menu. To do this we press the key.

Figure 7 The result of all of our efforts is the graph shown in Figure 7. Note that the inequality is TRUE, that is tester() produces a 1, for all values of x greater than or equal to 1. For values of x that make the inequality FALSE, the graph is at level 0, which, unfortunately, means that the graph is right on top of the x-axis.
We know from solving the problem by hand that the answer is indeed all values of x greater than or equal to 1. However, looking at the graph in Figure 7 it is not quite so easy to be so exact. The raised part of the graph does not start at exactly 1. What other tools are there to help us examine the graph?

Figure 8 To move to Figure 8 we have pressed the key to place the calculator in TRACE mode. The initial x value is .12658228, and the corresponding y value is 0, indicating that the inequality is FALSE for that x value.

Figure 9 We can use the cursor key, , to move the pointer to the right. Figure 9 captures the screen with the pointer as far to the right as possible before it jumps to the raised portion of the graph.

Figure 10 Pressing the key one more time moves the pointer to x=1.1392409 with the corresponding y value 1, indicating that the inequality is TRUE for that x value.

Figure 11 The x values chosen by the calculator in Figures 8, 9, and 10 have been inconvenient. We can have the calculator choose "nicer" x values by using the ZOOM window to change the WINDOW settings. Therefore, in Figure 11, we return to the ZOOM menu via the key. This time we will choose the ZoomDec option, via the key, to set the WINDOW so that we have nice decimal x values.

Figure 12 The actions described above produce the graph shown in Figure 12. Note that both the x and the y scales have changed. The raised part of the graph, the values of x that make the inequality TRUE, remain at the 1 level, but 1 appears further away from the x-axis.
We can return to the TRACE mode by pressing . Then we can use the key to move the pointer to x=.9 as shown in Figure 13.

Figure 13 The x value is as expected, 0.9, and the corresponding y value is 0.

Figure 14 Pressing once moves the pointer to 1.1, where we have a TRUE values. Note that in the ZoomDec setting, pressing the the right arrow, or the left arrow, cursor key results in a change of the x-value by 0.2. later, in Figures 19 and 20, we will change that setting. At the moment, we can demonstrate another feature of the TI-89. In particular, although the cursor keys allow us to look at certain values, we can enter any x-value that we desire, and the calculator will both determine the appropriate y-value.

Figure 15 To move from Figure 14 to Figure 15 we have pressed the key. This will allow us to ask the calculator to evaluate the function when x=1 and to move the pointer on the screen to the pixel that corresponds to that value. Once we press the key the calculator does the evaluation and moves the pointer, as is shown in Figure 16.

Figure 16 From Figure 16 we can determine that x=1 is part of the solution to the original inequality.

Figure 17 Let us test a value just less than 1. In particular, for Figure 17 we have pressed the key2. We are getting ready to ask the calculator to evaluate the function and plot the result for x=.99.

Figure 18 We press to accept the .99 value. The calculator shows, in Figure 18, that the inequality is FALSE (produces the value 0) when x=.99.

Figure 19 In all of the images in Figures 12 through 18, a careful observer might be concerned by the "step-like" climb from FALSE to TRUE. An enlargement of the critical area shows these steps. The explanation for these steps is as follows. For all of these images, the ZoomDec setting has caused the xres value to be set at 2. The calculator has evaluated every second x-value starting with xMin and with steps set at 0.1. Therefore, the calculator evaluated x at 0.9 and at 1.1, in fact we observered exactly those values in Figures 13 and 14. The calcualtor also tries to vertically connect adjacent plotted points. Therefore, the calculator had to connect (0.9, 0) with the point (1.1, 1). In doing this the calculator has to move from the x-axis up to the level of the y-value 1. This creates the steps seen in the Figures above. The same kind of step can be seen in Figure 7, but it is not quite as prominent because the change from the x-axis to the y-value 1 is so small in that image.
To further demonstrate this, open the WINDOW screen by pressing , as shown in Figure 19. Note the value for xres at the bottom.

Figure 20 In Figure 20 we have used the key to move down to the xres line and we have changed the value to 1 by pressing the key. To return to the graph window we press . The result is shown in Figure 21.

Figure 21 The new graph in Figure 21 still has a step, but it is the single step caused by trying to connect the point (0.9, 0) to the point (1.0, 1).
We can make the graph a bit easier to follow if we change the function so that FALSE corresponds to something other than 0. This will get the FALSE portion of the graph off the x-axis.

Figure 22 In Figure 22 we return to the "y=" screen via the keys. The "y1=" line is highlighted.

Figure 23 We press to move the highlighted function of Figure 22 down to the function entry and edit line, as shown in Figure 23.

Figure 24 Just pressing the key will move the cursor to the right end of the function definition (because the entire function was highlighted in Figure 23). Then we append the "*4–2" by pressing the keys. This produces the line at the bottom of Figure 24. Multiplying by 4 and then subtracting 2 means that a TRUE value will now produce 2 (1*4–2) and a FALSE value will now produce ^–2 (0*4–2).

Figure 25 To accept that line (the one at the bottom of Figure 24), we press the key. Figure 25 has the new function definition assigned to "y1=" even though the additional characters can not be seen on that screen.
Then we press to move to Figure 26 with a graph of the new function.

Figure 26 The graph shown in Figure 26 has the advantage that we can see the FALSE region as well as the TRUE region. In addition, because xres is still 1, the calculator is plotting each of the x-values. Therefore, it connects the point (0.9, ^–2) to the next point (1.0, 2).

Figure 1	We have opened the y= menu in Figure 1 by pressing the keys. The calculator used to generate these screens did not have any previously defined graphs on it. Had a graph been defined earlier on the calculator, we would need to CLEAR any old definitions in order to arrive at Figure 1.
Figure 2	We need to from the top portion of the screen in Figure 1 to the function entry and edit line at the bottom. To do this we can press the key. Now we are ready to enter the inequality.
Figure 3	The actual command that we want to enter is tester("3(x+2)–5x4x",x) We can generate the tester by finding it in the VAR-LINK menu, or we could just presss the keys to lock the calculator in alphabetic mode, and then press , followed by to leave alphabetic mode. Then we can continue creating the line shown in Figure 3 via the keys . Now we need to genreate the "less than or equal to" sign. It is not on the keyboard, but we can get it by pressing the keys. And, we complete Figure 3 with the key. The desired command is not done. More text must be entered, and we will do that in Figure 4.
Figure 4	To complete the command started in Figure 3 we press and . The result should be the line shown at the bottom of Figure 4.
Figure 5	Move from Figure 4 to Figure 5 by pressing the key. The calculator is now ready for a second function. We do not have a second function to graph. Instead, we wish to graph the first function on a standard graph. We will move to the ZOOM window to select such a setting. Note the in the menu at the top of the screen.
Figure 6	We open the ZOOM window by pressing the key. The ZOOM window is shown in Figure 6. We want to select the ZoomStd item, item 6, from the menu. To do this we press the key.
Figure 7	The result of all of our efforts is the graph shown in Figure 7. Note that the inequality is TRUE, that is tester() produces a 1, for all values of x greater than or equal to 1. For values of x that make the inequality FALSE, the graph is at level 0, which, unfortunately, means that the graph is right on top of the x-axis. We know from solving the problem by hand that the answer is indeed all values of x greater than or equal to 1. However, looking at the graph in Figure 7 it is not quite so easy to be so exact. The raised part of the graph does not start at exactly 1. What other tools are there to help us examine the graph?
Figure 8	To move to Figure 8 we have pressed the key to place the calculator in TRACE mode. The initial x value is .12658228, and the corresponding y value is 0, indicating that the inequality is FALSE for that x value.
Figure 9	We can use the cursor key, , to move the pointer to the right. Figure 9 captures the screen with the pointer as far to the right as possible before it jumps to the raised portion of the graph.
Figure 10	Pressing the key one more time moves the pointer to x=1.1392409 with the corresponding y value 1, indicating that the inequality is TRUE for that x value.
Figure 11	The x values chosen by the calculator in Figures 8, 9, and 10 have been inconvenient. We can have the calculator choose "nicer" x values by using the ZOOM window to change the WINDOW settings. Therefore, in Figure 11, we return to the ZOOM menu via the key. This time we will choose the ZoomDec option, via the key, to set the WINDOW so that we have nice decimal x values.
Figure 12	The actions described above produce the graph shown in Figure 12. Note that both the x and the y scales have changed. The raised part of the graph, the values of x that make the inequality TRUE, remain at the 1 level, but 1 appears further away from the x-axis. We can return to the TRACE mode by pressing . Then we can use the key to move the pointer to x=.9 as shown in Figure 13.
Figure 13	The x value is as expected, 0.9, and the corresponding y value is 0.
Figure 14	Pressing once moves the pointer to 1.1, where we have a TRUE values. Note that in the ZoomDec setting, pressing the the right arrow, or the left arrow, cursor key results in a change of the x-value by 0.2. later, in Figures 19 and 20, we will change that setting. At the moment, we can demonstrate another feature of the TI-89. In particular, although the cursor keys allow us to look at certain values, we can enter any x-value that we desire, and the calculator will both determine the appropriate y-value.
Figure 15	To move from Figure 14 to Figure 15 we have pressed the key. This will allow us to ask the calculator to evaluate the function when x=1 and to move the pointer on the screen to the pixel that corresponds to that value. Once we press the key the calculator does the evaluation and moves the pointer, as is shown in Figure 16.
Figure 16	From Figure 16 we can determine that x=1 is part of the solution to the original inequality.
Figure 17	Let us test a value just less than 1. In particular, for Figure 17 we have pressed the key2. We are getting ready to ask the calculator to evaluate the function and plot the result for x=.99.
Figure 18	We press to accept the .99 value. The calculator shows, in Figure 18, that the inequality is FALSE (produces the value 0) when x=.99.
Figure 19	In all of the images in Figures 12 through 18, a careful observer might be concerned by the "step-like" climb from FALSE to TRUE. An enlargement of the critical area shows these steps. The explanation for these steps is as follows. For all of these images, the ZoomDec setting has caused the xres value to be set at 2. The calculator has evaluated every second x-value starting with xMin and with steps set at 0.1. Therefore, the calculator evaluated x at 0.9 and at 1.1, in fact we observered exactly those values in Figures 13 and 14. The calcualtor also tries to vertically connect adjacent plotted points. Therefore, the calculator had to connect (0.9, 0) with the point (1.1, 1). In doing this the calculator has to move from the x-axis up to the level of the y-value 1. This creates the steps seen in the Figures above. The same kind of step can be seen in Figure 7, but it is not quite as prominent because the change from the x-axis to the y-value 1 is so small in that image. To further demonstrate this, open the WINDOW screen by pressing , as shown in Figure 19. Note the value for xres at the bottom.
Figure 20	In Figure 20 we have used the key to move down to the xres line and we have changed the value to 1 by pressing the key. To return to the graph window we press . The result is shown in Figure 21.
Figure 21	The new graph in Figure 21 still has a step, but it is the single step caused by trying to connect the point (0.9, 0) to the point (1.0, 1). We can make the graph a bit easier to follow if we change the function so that FALSE corresponds to something other than 0. This will get the FALSE portion of the graph off the x-axis.
Figure 22	In Figure 22 we return to the "y=" screen via the keys. The "y1=" line is highlighted.
Figure 23	We press to move the highlighted function of Figure 22 down to the function entry and edit line, as shown in Figure 23.
Figure 24	Just pressing the key will move the cursor to the right end of the function definition (because the entire function was highlighted in Figure 23). Then we append the "4–2" by pressing the keys. This produces the line at the bottom of Figure 24. Multiplying by 4 and then subtracting 2 means that a TRUE* value will now produce 2 (14–2) and a FALSE* value will now produce ^–2 (0*4–2).
Figure 25	To accept that line (the one at the bottom of Figure 24), we press the key. Figure 25 has the new function definition assigned to "y1=" even though the additional characters can not be seen on that screen. Then we press to move to Figure 26 with a graph of the new function.
Figure 26	The graph shown in Figure 26 has the advantage that we can see the FALSE region as well as the TRUE region. In addition, because xres is still 1, the calculator is plotting each of the x-values. Therefore, it connects the point (0.9, ^–2) to the next point (1.0, 2).