Graphing Parametric Equations on the TI-86

The TI-86 allows us to graph parametric equations. This page demonstrates two examples of graphing parametric equations. However, before we even start that process, we will us the first six Figures to quickly review the steps used in graphing functions. We will obtain the graph of the function

y = (x–4)(x+2)/4

Figure 1 We start the process by looking at the MODE screen. On the TI-86 we do this by pressing the keys. Note that the highlight, the cursor, in the first line is normally blinking. The image in Figure 1 was captured with that blinking cursor covering the selection "Normal".
The fifth line in Figure 1 is of particular importance. The setting shown in Figure 1 indicates that the calculator is set to plot Functions.

Figure 2 We can leave Figure 1 by pressing the key. On this calculator the display shifts to that shown in Figure 2. We have yet to define a function. Therefore, the calculator has no functions to graph.

Figure 2a We move to the y(x)= screen, via the key. Here we define our function by it as shown in Figure 2a.
Figure 3 The key sequence will perform the WIND option fromthe top menu of Figure 2a. This will open the WINDOW screen shown in Figure 3. Note that these are the default values for obtaining a "decimal" display. That is, with these settings, the TI-86 will associate each pixel with coordinates that have values that change by exactly 0.1 at each step.

Figure 4 Finally, we press to produce the graph shown in Figure 4.

Figure 5 We can leave Figure 4 and move to Figure 5 by pressing the key to shift into TRACE mode. In Figure 5 we can find the trace indicator at the very bottom of the gaph, at the coordinates (0,^–2).

Figure 6 By pressing the key 15 times we move the trace indicator to the position shown in Figure 6. At that point x is ^–1.5 and y is ^–0.6875.

So much for the review of graphing functions. Now we want to look at the steps needed to graph parametric equations. We will use the following as our example:

x = t + 4
y = (t² + 6t)/4 Note that the first equation gives x as function of t, and the second equation gives y as a function of t. Therefore, if we choose a value for t, these two equations produce an x value and a y value. For example, if we were to let t be ^–5.5, then the first equation would mean that x would be ^–1.5, while the second equation would mean that y would be ^–0.6875.

In order to have the TI-86 graph parametric equations we need to shift it into parametric mode.

Figure 7 We return to the MODE screen by pressing the keys. THen we move down to the fifth line by using the key four times. Next we move the blinking cursor to the right by pressing the key twice. And, finally, we actually select Parametric mode by pressing the key. The result is as shown in Figure 7.

Figure 8 We use the key to open the GRAPH menu shown in Figure 8. Note that the first menu item has been changed. Where we used to find the y(x)= option, we now find the E(t)= option.

Figure 9 Pressing to select that E(t)= option, the calculator displays the screen that prompts us for the two parametric equations. The first equation needs to give x as a function of t, while the second equation needs to give y as a function of t.

Figure 10 In Figure 10 we have entered the parametric equations for our problem. Note that we generate the t in the equations by using the key to select the t from the sub-menu.

Figure 11 Next, we will check out the settings for the WINDow screen. We press to select WIND from the top menu of Figure 10. This produces Figure 11, where the blinking cursor is covering the value 0.
This screen, the WINDOW screen for parametric mode, has three new values in it, tMin, tMax, and tStep. These settings will be used to create the values of t that will be used to plot our equations.
You might notice that although tMax is not a nice even value, it is not just a random value. Rather, tMax is twice the value of . tStep is set so that there are 49 values of t to be plotted, from tMin through tMax.

Figure 12 Figure 11 displayed the first six values in the parametric equation WINDOW screen. We can use the key to scroll down so that we can see the remaining 3 values on that screen.

Figure 13 We press to actually graph the equations. The TI-86 will generate each of the 49 values of t from tMin through tMax. For each value of t, the calculator determines the appropriate value of x and y from the parametric equations. Then, the calcualtor graphs that (x,y) point on the screen.
When t is 0 we will get x=4 and y=0. Therefore, the first point that is graphed is the point (4,0).

Figure 14 We can see from Figure 13 that we need to change the values that will be assigned to t. We press the key to return tot he WINDOW screen. Then, in Figure 14, we have changed the values of tMin, tMax, and tStep, so that t will vary from ^–10 to 10 in steps of one tenth.

Figure 15 We press to initiate a new graph, based on the new values for t. The new graph appears in Figure 15.

Figure 16 Pressing shifts the calculator into TRACE mode. Note that the calculator displays the values of t, x, and y at the bottom of the screen. We start at the lowest value of t, that is at tMin.

Figure 17 The calcualtor allows us to trace the graph by changing the value of t. Each time we move the cursor left or right, the calculator changes t according to the value in tStep and within the limits of tMin and tMax. For example, in Figure 17, we have used the key to move the cursor to the right until t has the value ^–5.5, which means that x=^–1.5 and y=^–0.6875.

Figure 18 The graph in Figure 17 pretty much filled the calculator image. Let us return to the WINDOW settings and make some changes to enlarge the range of the values that we will display. Figure 17 was in TRACE mode. We press to leave TRACE mode and restore the menu. Then press to select the WIND option. This will display the various WINDOW settings. For Figure 18 we have used the key to move down that list of values, changing the xMax, xScl, yMax, and yScl values as shown.

Figure 19 We press to generate a new graph, based on the changed WINDOW settings. That graph, shown in Figure 19 demonstrates the limits on the graph imposed by having t take on values between ^– 10 and 10.

The function graph in Figure 6 looks remarkabley like the parametric equation graph of Figure 17. In fact, these are the same relations. If westart with our function

y = (x – 4)(x + 2)/4 and we let t = x – 4 then we know that t + 6 = x + 2 and, therefore, x = t + 4 and y = t(t+6)/4 which are the parametric equations that we used.

That first example allowed us to convert a function to parametric equations. However, this does not demonstrate the power and flexibility of parametric equations. To do this, consider the parametric equations

x = (t–3)t(t+4)/15 – 5
y = (t–1)(t+2)/4 – 4 We will graph those equations on the TI-86.

Figure 20 We return to the E(t)= screen by pressing the key. Then we enter the equations to replace the ones that were using earlier. Figure 20 reflects the new equations.

Figure 21 Before we graph these equations we press to open the ZOOM sub-menu. This is shown in Figure 21.

Figure 22 We press to select the ZSTD option from the sub-menu in Figure 21. That option will set the standard values into the various WINDOW settings, and it will produce the graph of the parametric equations. That graph is shown in Figure 22.
We immediately note that the graph in Figure 22 is NOT a function. It fails the vertical line test. In addition, given the standard settings, we are not sure just what the rest of the graph should look like. We need to go back and modify those settings.

Figure 23 We return to the WINDOW screen by pressing the key. Figure 23 shows the standard settings for parametric equations.

Figure 24 Figure 24 shows the WINDOW settings after we have made some modifications. We will let t run from ^– 5 to 4 in steps of 0.1.

Figure 25 To start Figure 25 we press the key. Doing so will actually generate Figure 27. However, in Figure 25 we have caught the graph in progress. In this fashion we can see that the graph starts at about (^–7.7,0.5) and it sweeps to the right and down, only to turn back to the left at about x=^–3.5.

Figure 26 Figure 26 represents a further step in the graph as it moves to completion in Figure 27. Note that the horizontal motion has changed back to moving toward the right. And, we are no longer seeing the graph go down; now it is moving up.

Figure 27 Figure 27 has the complete graph. Complete, that is, within the limits that we imposed on t back in Figure 24. If we change those values, we will change the graph.

Figure 28 In Figure 28 we have returned to the WINDOW settings by pressing . In addition, we have altered the setting for tMax to be 6.

Figure 29 Press to draw a new graph, shown in Figure 29.

Figure 30 As before, we press the key to move into TRACE mode. Note that the calculator displays the appropriate values of t, x, and y.

Figure 31 Now that we are in TRACE mode, we press the key to move the trace pointer to the next value of t. In Figure 31 we have pressed that key about 20 times to move the pointer to near the right edge of the loop.

Figure 32 If we continue to press the key, the trace indicator continues to follow the graph, just as we saw it being drawn back in Figures 25 through 27. Note that t continues to increase, even though x and y may be increasing or decreasing.

Figure 33 Figure 33 merely continues the demonstration of the TRACE mode, using the key to move the pointer to the next value of t.

Figure 1	We start the process by looking at the MODE screen. On the TI-86 we do this by pressing the keys. Note that the highlight, the cursor, in the first line is normally blinking. The image in Figure 1 was captured with that blinking cursor covering the selection "Normal". The fifth line in Figure 1 is of particular importance. The setting shown in Figure 1 indicates that the calculator is set to plot Functions.
Figure 2	We can leave Figure 1 by pressing the key. On this calculator the display shifts to that shown in Figure 2. We have yet to define a function. Therefore, the calculator has no functions to graph.
Figure 2a	We move to the y(x)= screen, via the key. Here we define our function by it as shown in Figure 2a.
Figure 3	The key sequence will perform the WIND option fromthe top menu of Figure 2a. This will open the WINDOW screen shown in Figure 3. Note that these are the default values for obtaining a "decimal" display. That is, with these settings, the TI-86 will associate each pixel with coordinates that have values that change by exactly 0.1 at each step.
Figure 4	Finally, we press to produce the graph shown in Figure 4.
Figure 5	We can leave Figure 4 and move to Figure 5 by pressing the key to shift into TRACE mode. In Figure 5 we can find the trace indicator at the very bottom of the gaph, at the coordinates (0,^–2).
Figure 6	By pressing the key 15 times we move the trace indicator to the position shown in Figure 6. At that point x is ^–1.5 and y is ^–0.6875.

Figure 7	We return to the MODE screen by pressing the keys. THen we move down to the fifth line by using the key four times. Next we move the blinking cursor to the right by pressing the key twice. And, finally, we actually select Parametric mode by pressing the key. The result is as shown in Figure 7.
Figure 8	We use the key to open the GRAPH menu shown in Figure 8. Note that the first menu item has been changed. Where we used to find the y(x)= option, we now find the E(t)= option.
Figure 9	Pressing to select that E(t)= option, the calculator displays the screen that prompts us for the two parametric equations. The first equation needs to give x as a function of t, while the second equation needs to give y as a function of t.
Figure 10	In Figure 10 we have entered the parametric equations for our problem. Note that we generate the t in the equations by using the key to select the t from the sub-menu.
Figure 11	Next, we will check out the settings for the WINDow screen. We press to select WIND from the top menu of Figure 10. This produces Figure 11, where the blinking cursor is covering the value 0. This screen, the WINDOW screen for parametric mode, has three new values in it, tMin, tMax, and tStep. These settings will be used to create the values of t that will be used to plot our equations. You might notice that although tMax is not a nice even value, it is not just a random value. Rather, tMax is twice the value of . tStep is set so that there are 49 values of t to be plotted, from tMin through tMax.
Figure 12	Figure 11 displayed the first six values in the parametric equation WINDOW screen. We can use the key to scroll down so that we can see the remaining 3 values on that screen.
Figure 13	We press to actually graph the equations. The TI-86 will generate each of the 49 values of t from tMin through tMax. For each value of t, the calculator determines the appropriate value of x and y from the parametric equations. Then, the calcualtor graphs that (x,y) point on the screen. When t is 0 we will get x=4 and y=0. Therefore, the first point that is graphed is the point (4,0).
Figure 14	We can see from Figure 13 that we need to change the values that will be assigned to t. We press the key to return tot he WINDOW screen. Then, in Figure 14, we have changed the values of tMin, tMax, and tStep, so that t will vary from ^–10 to 10 in steps of one tenth.
Figure 15	We press to initiate a new graph, based on the new values for t. The new graph appears in Figure 15.
Figure 16	Pressing shifts the calculator into TRACE mode. Note that the calculator displays the values of t, x, and y at the bottom of the screen. We start at the lowest value of t, that is at tMin.
Figure 17	The calcualtor allows us to trace the graph by changing the value of t. Each time we move the cursor left or right, the calculator changes t according to the value in tStep and within the limits of tMin and tMax. For example, in Figure 17, we have used the key to move the cursor to the right until t has the value ^–5.5, which means that x=^–1.5 and y=^–0.6875.
Figure 18	The graph in Figure 17 pretty much filled the calculator image. Let us return to the WINDOW settings and make some changes to enlarge the range of the values that we will display. Figure 17 was in TRACE mode. We press to leave TRACE mode and restore the menu. Then press to select the WIND option. This will display the various WINDOW settings. For Figure 18 we have used the key to move down that list of values, changing the xMax, xScl, yMax, and yScl values as shown.
Figure 19	We press to generate a new graph, based on the changed WINDOW settings. That graph, shown in Figure 19 demonstrates the limits on the graph imposed by having t take on values between ^– 10 and 10.

Figure 20	We return to the E(t)= screen by pressing the key. Then we enter the equations to replace the ones that were using earlier. Figure 20 reflects the new equations.
Figure 21	Before we graph these equations we press to open the ZOOM sub-menu. This is shown in Figure 21.
Figure 22	We press to select the ZSTD option from the sub-menu in Figure 21. That option will set the standard values into the various WINDOW settings, and it will produce the graph of the parametric equations. That graph is shown in Figure 22. We immediately note that the graph in Figure 22 is NOT a function. It fails the vertical line test. In addition, given the standard settings, we are not sure just what the rest of the graph should look like. We need to go back and modify those settings.
Figure 23	We return to the WINDOW screen by pressing the key. Figure 23 shows the standard settings for parametric equations.
Figure 24	Figure 24 shows the WINDOW settings after we have made some modifications. We will let t run from ^– 5 to 4 in steps of 0.1.
Figure 25	To start Figure 25 we press the key. Doing so will actually generate Figure 27. However, in Figure 25 we have caught the graph in progress. In this fashion we can see that the graph starts at about (^–7.7,0.5) and it sweeps to the right and down, only to turn back to the left at about x=^–3.5.
Figure 26	Figure 26 represents a further step in the graph as it moves to completion in Figure 27. Note that the horizontal motion has changed back to moving toward the right. And, we are no longer seeing the graph go down; now it is moving up.
Figure 27	Figure 27 has the complete graph. Complete, that is, within the limits that we imposed on t back in Figure 24. If we change those values, we will change the graph.
Figure 28	In Figure 28 we have returned to the WINDOW settings by pressing . In addition, we have altered the setting for tMax to be 6.
Figure 29	Press to draw a new graph, shown in Figure 29.
Figure 30	As before, we press the key to move into TRACE mode. Note that the calculator displays the appropriate values of t, x, and y.
Figure 31	Now that we are in TRACE mode, we press the key to move the trace pointer to the next value of t. In Figure 31 we have pressed that key about 20 times to move the pointer to near the right edge of the loop.
Figure 32	If we continue to press the key, the trace indicator continues to follow the graph, just as we saw it being drawn back in Figures 25 through 27. Note that t continues to increase, even though x and y may be increasing or decreasing.
Figure 33	Figure 33 merely continues the demonstration of the TRACE mode, using the key to move the pointer to the next value of t.