Points of Intersection on the TI-86

Example 13 in Chapter 2 Section 4 in the text contains a calculator generated graph of two functions: f(x) = ^–2x + 2
g(x) = 2x² – 4x –2 Earlier, as part of the supporting material for Example 14 from Chapter 1 section 2, the book and the web page 128601.htm demonstrated the use of the ISECT operation for locating the point of intersection for two functions. We continue that demonstration here, using the two functions from Example 13.

Please note that the calculator used to produce the graphs on this page had just been used to produce the graphs and screen images from the web page 238601.htm. As a result there is some residual material on this calculator.

Figure 1 Figure 1 is taken from the y(x)= screen on a TI-86 after we have defined the two functions. The first function is a straight line, and the second is a quadratic which will be graphed as a parabola.

Figure 2 Pressing moves the display to Figure 2. Here we see some unexpected items. For one thing, the graph shows a number of points that are not related to this problem at all. Second, there are two straight lines that have been graphed here, only one of which has slope equal to ^–2. Third, the graph of the parabola is made up of separated points. And, fourth, the WINDOW settings for the graph are not obvious.

Figure 3 We will use the ZOOM feature to change the WINDOW settings to the standard values. To do this we press to open the sub-menu shown in Figure 3. Then, to select ZSTD we press . This will produce Figure 4.

Figure 4 In Figure 4 we have the more usual Window settings, namely both x and y range between ^–10 and 10. In addition, the second straight line from Figures 2 and 3, the line that approximated the plotted points, has disappeared. That line was the drawing of the regression equation computed from those points. It was left over from earlier work on this calculator. On the TI-86 we can obtain that regresssion equation, after performing the LinR analysis, by asking the calculator to DRREG, DRaw the REGression equation. However, that drawing disappears as soon as we have the calculator recreate a graph. Changing the WINDOW settings via the ZSTD option forces the TI-86 to reconstruct the graph.
We still have the points plotted, although we can only see two of them within the given range, and the parabola is made up of disconnected dots.

Figure 5 To make the parabola appear as connected points, we will close the submenu, via the key, and then use the key to shift the main menu display to the next set of options, shown in Figure 5. From that set of options we select FORMT by pressing the key. This will open the options screen shown in Figure 6.

Figure 6 The DrawDot option needs to be changed. We use the to move the highlight down to the DrawLine option and then press to selct that option. The result is shown in Figure 7.

Figure 7 Now that the option is set as we want it, we press to redraw the graph.

Figure 8 Here the parabola is connected, but we still have those plotted points left over from some earlier work. To clear those points we need to turn off the statistical plot. To do that we need to return to the STAT menu.

Figure 9 We press to open the STAT menu shown in Figure 9. From that menu we press to choose PLOT. That will open the window in Figure 10.

Figure 10 Now we can see that Plot1 is indeed On. The easiest remedy will be to turn off all of the plots. We do this by using the key to select PlOff. That option causes the PlOff command to be pasted onto the base window in Figure 11.

Figure 11 Once the command has been pasted onto the window, we press to perform it. The calcualtor responds with Done.
Then, we return to the graph by pressing .

Figure 12 In Figure 12 we finally see the graph without the distracting plotted points.
Our purpose here is to use the ISECT command to find the two points of intersection. We could modify the WINDOW settings to have the graph come close to the one presented in the text, but that will not be necessary.

Figure 13 To get to the ISECT option we need to use the key to make the MATH option appear in the main menu, and then use to open the MATH sub-menu. The result is shown in Figure 13. ISECT is still a step away. We will need to press to see more options in the MATH sub-menu.

Figure 14 We find ISECT as the third item in the sub-menu. Press to select that option.

Figure 15 ISECT goes through a number of steps. First, we need to choose the two curves to use. It is possible that we would have more than 2 curves graphed at the same time. The calculator wants us to choose the two graphs to use. This seems a bit superfluous here given that there are only two functions on this graph. However, we will have to play along with the calcualtor.
The TI-86 is proposing the first function as one of the two curves. We press to accept that choice.

Figure 16 In Figure 16 the TI-86 proposes the second function as the the second curve. Again, press to accept that choice.

Figure 17 Now the calculator is asking for a Guess, and it is proposing the point (0,^–2) as such a guess. We want to be sure that the calculator finds the left point of intersection. Therefore, we will improve the guess to help the calculator. We do this by pressing the key to move the Guess closer to the point of intersection.

Figure 18 In Figure 18 we have a Guess that is reasonably close to the left point of intersection. We press to accept that point and have the calculator do the rest of the work.

Figure 19 The TI-86 has determined that the left point of intersection is (^–1,4).
Now we want to obtain the coordinates of the other point of intersection. To do this we step through the same sequence of screens, but changing our Guess so that it is close to the right point of intersection.

Figure 20 We press to leave Figure 19. This will return us to the main graph menu. There choose the MATH option, use the MORE key to display new options in the sub-menu, and select ISECT. The calculator proposes the first curve as one of the two curves to use. We use the key to accept that choice.

Figure 21 The calculator proposes the second curve as the other curve to use. We use the key to accept that choice.

Figure 22 The TI-86 proposes a point to use as the Guess. We do not want to use that guess because it is too close to the left point of intersection. Therefore, we will use the to shift the guess close to the right point of intersection.

Figure 23 In Figure 23 the new Guess point has been set. Press to accept that value.

Figure 24 ISECT has done its work. The right point of intersection is identified as (2,^–2).

Figure 1	Figure 1 is taken from the y(x)= screen on a TI-86 after we have defined the two functions. The first function is a straight line, and the second is a quadratic which will be graphed as a parabola.
Figure 2	Pressing moves the display to Figure 2. Here we see some unexpected items. For one thing, the graph shows a number of points that are not related to this problem at all. Second, there are two straight lines that have been graphed here, only one of which has slope equal to ^–2. Third, the graph of the parabola is made up of separated points. And, fourth, the WINDOW settings for the graph are not obvious.
Figure 3	We will use the ZOOM feature to change the WINDOW settings to the standard values. To do this we press to open the sub-menu shown in Figure 3. Then, to select ZSTD we press . This will produce Figure 4.
Figure 4	In Figure 4 we have the more usual Window settings, namely both x and y range between ^–10 and 10. In addition, the second straight line from Figures 2 and 3, the line that approximated the plotted points, has disappeared. That line was the drawing of the regression equation computed from those points. It was left over from earlier work on this calculator. On the TI-86 we can obtain that regresssion equation, after performing the LinR analysis, by asking the calculator to DRREG, DRaw the REGression equation. However, that drawing disappears as soon as we have the calculator recreate a graph. Changing the WINDOW settings via the ZSTD option forces the TI-86 to reconstruct the graph. We still have the points plotted, although we can only see two of them within the given range, and the parabola is made up of disconnected dots.
Figure 5	To make the parabola appear as connected points, we will close the submenu, via the key, and then use the key to shift the main menu display to the next set of options, shown in Figure 5. From that set of options we select FORMT by pressing the key. This will open the options screen shown in Figure 6.
Figure 6	The DrawDot option needs to be changed. We use the to move the highlight down to the DrawLine option and then press to selct that option. The result is shown in Figure 7.
Figure 7	Now that the option is set as we want it, we press to redraw the graph.
Figure 8	Here the parabola is connected, but we still have those plotted points left over from some earlier work. To clear those points we need to turn off the statistical plot. To do that we need to return to the STAT menu.
Figure 9	We press to open the STAT menu shown in Figure 9. From that menu we press to choose PLOT. That will open the window in Figure 10.
Figure 10	Now we can see that Plot1 is indeed On. The easiest remedy will be to turn off all of the plots. We do this by using the key to select PlOff. That option causes the PlOff command to be pasted onto the base window in Figure 11.
Figure 11	Once the command has been pasted onto the window, we press to perform it. The calcualtor responds with Done. Then, we return to the graph by pressing .
Figure 12	In Figure 12 we finally see the graph without the distracting plotted points. Our purpose here is to use the ISECT command to find the two points of intersection. We could modify the WINDOW settings to have the graph come close to the one presented in the text, but that will not be necessary.
Figure 13	To get to the ISECT option we need to use the key to make the MATH option appear in the main menu, and then use to open the MATH sub-menu. The result is shown in Figure 13. ISECT is still a step away. We will need to press to see more options in the MATH sub-menu.
Figure 14	We find ISECT as the third item in the sub-menu. Press to select that option.
Figure 15	ISECT goes through a number of steps. First, we need to choose the two curves to use. It is possible that we would have more than 2 curves graphed at the same time. The calculator wants us to choose the two graphs to use. This seems a bit superfluous here given that there are only two functions on this graph. However, we will have to play along with the calcualtor. The TI-86 is proposing the first function as one of the two curves. We press to accept that choice.
Figure 16	In Figure 16 the TI-86 proposes the second function as the the second curve. Again, press to accept that choice.
Figure 17	Now the calculator is asking for a Guess, and it is proposing the point (0,^–2) as such a guess. We want to be sure that the calculator finds the left point of intersection. Therefore, we will improve the guess to help the calculator. We do this by pressing the key to move the Guess closer to the point of intersection.
Figure 18	In Figure 18 we have a Guess that is reasonably close to the left point of intersection. We press to accept that point and have the calculator do the rest of the work.
Figure 19	The TI-86 has determined that the left point of intersection is (^–1,4). Now we want to obtain the coordinates of the other point of intersection. To do this we step through the same sequence of screens, but changing our Guess so that it is close to the right point of intersection.
Figure 20	We press to leave Figure 19. This will return us to the main graph menu. There choose the MATH option, use the MORE key to display new options in the sub-menu, and select ISECT. The calculator proposes the first curve as one of the two curves to use. We use the key to accept that choice.
Figure 21	The calculator proposes the second curve as the other curve to use. We use the key to accept that choice.
Figure 22	The TI-86 proposes a point to use as the Guess. We do not want to use that guess because it is too close to the left point of intersection. Therefore, we will use the to shift the guess close to the right point of intersection.
Figure 23	In Figure 23 the new Guess point has been set. Press to accept that value.
Figure 24	ISECT has done its work. The right point of intersection is identified as (2,^–2).