Points of Intersection on the TI-89

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 128901.htm demonstrated the use of the Intersection 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 238901.htm. As a result there is some residual material on this calculator.

Figure 1 Figure 1 is taken from the y= screen on a TI-89 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, 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 window shown in Figure 3. Then, to select ZoomStd 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. Two of the plotted points remain visible on the new WINDOW. They are left over from earlier work on this calculator.

Figure 5 To generate Figure 5 we need to return to the y= screen. We can press to do this. On this screen we note that Plot 1 is not only defined, but it is active, as noted by the check mark to the left of Plot 1. Furthermore, we have used the key to move the highlight to Plot 1.

Figure 6 We will unselect Plot 1 by pressing the key. Note that the check mark has been removed. The plot remains defined, but it will not be graphed.

Figure 7 returns the calculator to the graph screen shown in Figure 7.
Our purpose here is to use the Intersection 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.
Then, use to select the Math tab at the top of the window.

Figure 8 From the options shown in Figure 8, we want to choose item 5: Intersection. Press to make that choice.

Figure 9 Iintersection 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-89 is proposing the first function as one of the two curves. We press to accept that choice.

Figure 10 In Figure 10 the TI-89 proposes the second function as the the second curve. Again, press to accept that choice.

Figure 11 The TI-89 will look for a solution, a point of intersection, within a specified domain. In Figure 11, the TI-89 is proposing a left or Lower Bound for that specified domain. Assuming that we want to find the left point of intersection, the proposed point will not do. Therefore, we use the key to move the left or Lower Bound to the left of the point of intersection that we can see on the screen. That change is reflected in Figure 12.

Figure 12 The pointer on the screen is to the left of the desired point of intersection. We press to accept that lower bound.

Figure 13 The next task is to specify a bound to the right of the point of intersection. In Figure 13 the calculator is proposing a point that is on the wrong side of the intersection. We use the key to move the pointer to the right side of the intersection.

Figure 14 Now the pointer is to the right of the intersection. This gives an Upper Bound to the domain of values. We press to accept this point.
Before leaving Figure 14, please note that the TI-89 has left a small marker to indicate the location of the Lower Bound. That marker is obscured by the graph of the parabola. A better example will appear in Figure 18.

Figure 15 The TI-89 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 Lower Bound and Upper Bound so that they bracket the right point of intersection.

Figure 16 We press to open the Math menu. We press to select 5: Intersection. The calculator proposes the first curve. We press to accept that choice.

Figure 17 The calculator proposes the second curve. Again, accepts that choice.

Figure 18 The TI-89 proposed a Lower Bound. We pressed to accept that Lower Bound. That action left a marker on the screen to indicate the location of the Lower Bound. The TI-89 then proposed an Upper Bound. That proposed point is shown in Figure 18.

Figure 19 We use the key to position the proposed Upper Bound to the right of the other intersection. Then press to accept that Upper Bound.

Figure 20 Intersection has done its work. The right point of intersection is identified as (2,^–2).

Figure 21 The TI-89 is a powerful tool. The Figures given above demonstrated the use of the Intersection command for finding points of intersection on from two graphed functions. Those functions were given as f(x) = ^–2x + 2
g(x) = 2x² – 4x –2 We could just as easily have asked the calculator to solve ^–2x + 2 = 2x² – 4x –2 Figure 21 shows the command used to do exactly that.

Figure 22 Figure 22 gives the two values of x that solve the previous equation.

Figure 23 We can use the definitions of the two functions to find the corresponding y values.

Figure 1	Figure 1 is taken from the y= screen on a TI-89 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, 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 window shown in Figure 3. Then, to select ZoomStd 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. Two of the plotted points remain visible on the new WINDOW. They are left over from earlier work on this calculator.
Figure 5	To generate Figure 5 we need to return to the y= screen. We can press to do this. On this screen we note that Plot 1 is not only defined, but it is active, as noted by the check mark to the left of Plot 1. Furthermore, we have used the key to move the highlight to Plot 1.
Figure 6	We will unselect Plot 1 by pressing the key. Note that the check mark has been removed. The plot remains defined, but it will not be graphed.
Figure 7	returns the calculator to the graph screen shown in Figure 7. Our purpose here is to use the Intersection 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. Then, use to select the Math tab at the top of the window.
Figure 8	From the options shown in Figure 8, we want to choose item 5: Intersection. Press to make that choice.
Figure 9	Iintersection 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-89 is proposing the first function as one of the two curves. We press to accept that choice.
Figure 10	In Figure 10 the TI-89 proposes the second function as the the second curve. Again, press to accept that choice.
Figure 11	The TI-89 will look for a solution, a point of intersection, within a specified domain. In Figure 11, the TI-89 is proposing a left or Lower Bound for that specified domain. Assuming that we want to find the left point of intersection, the proposed point will not do. Therefore, we use the key to move the left or Lower Bound to the left of the point of intersection that we can see on the screen. That change is reflected in Figure 12.
Figure 12	The pointer on the screen is to the left of the desired point of intersection. We press to accept that lower bound.
Figure 13	The next task is to specify a bound to the right of the point of intersection. In Figure 13 the calculator is proposing a point that is on the wrong side of the intersection. We use the key to move the pointer to the right side of the intersection.
Figure 14	Now the pointer is to the right of the intersection. This gives an Upper Bound to the domain of values. We press to accept this point. Before leaving Figure 14, please note that the TI-89 has left a small marker to indicate the location of the Lower Bound. That marker is obscured by the graph of the parabola. A better example will appear in Figure 18.
Figure 15	The TI-89 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 Lower Bound and Upper Bound so that they bracket the right point of intersection.
Figure 16	We press to open the Math menu. We press to select 5: Intersection. The calculator proposes the first curve. We press to accept that choice.
Figure 17	The calculator proposes the second curve. Again, accepts that choice.
Figure 18	The TI-89 proposed a Lower Bound. We pressed to accept that Lower Bound. That action left a marker on the screen to indicate the location of the Lower Bound. The TI-89 then proposed an Upper Bound. That proposed point is shown in Figure 18.
Figure 19	We use the key to position the proposed Upper Bound to the right of the other intersection. Then press to accept that Upper Bound.
Figure 20	Intersection has done its work. The right point of intersection is identified as (2,^–2).
Figure 21	The TI-89 is a powerful tool. The Figures given above demonstrated the use of the Intersection command for finding points of intersection on from two graphed functions. Those functions were given as f(x) = ^–2x + 2 g(x) = 2x² – 4x –2 We could just as easily have asked the calculator to solve ^–2x + 2 = 2x² – 4x –2 Figure 21 shows the command used to do exactly that.
Figure 22	Figure 22 gives the two values of x that solve the previous equation.
Figure 23	We can use the definitions of the two functions to find the corresponding y values.