Graphing on the TI-86

Figure 1	We are assuming that the calculator is turned on and that there is nothing on the home screen. Figure 1 represents such a situation. The first thing that we will do is to move to the GRAPH screen by pressing the key.
Figure 2	Figure 2 represents a graph of some functions left over from an earlier user of the calculator. We want to get rid of these. Figures 3-7 demonstrate removing functions from the graph. If your calculator has no functions on it you may want to move directly to Figure 8, although it would be good to at least check to be sure that no functions are already defined.
Figure 3	We get to Figure 3 by pressing the key. This selects the first screen menu item, the item at the bottom of the keyscreen just above the key. That first menu item, y(x)=, brings up a display of all of the functions that are currently defined for graphing on the calculator. Each separate function starts with y index = such as y1= or y2=. If there is nothing after the equal sign then there is no function assigned there. In Figure 3 we see that there are 4 functions already assigned to y1 through y4. The blinking cursor starts on the first character of the first function, in this case a negative sign.
Figure 4	We can clear the first function by pressing the key. The result is shown in Figure 4. Note that the background black rectangle surrounding the equal sign in the first equation has disappeared. This is an indication that this equation has been de-selected.
Figure 5	To move to the next function, press the key. This puts the blinking cursor onto the negative sign in the second function, as shown in Figure 5.
Figure 6	We can clear the second function by pressing the key. The result is shown in Figure 6. Then we can move down to the next line with the key, clear that function with the key, move down again via another key, and clear that function with the key.
Figure 7	We leave Figure 6 and move to Figure 7 by pressing the key and then the key. That takes us to the GRAPH screen. Figure 7 represents the conclusion of our efforts to clear out all of the pre-existing functions. We now have a clear graph.

We can pick up here, assuming that no functions have been defined in the calculator. We force the calculator into graph mode by pressing the key. That should create a graph similar to Figure 7, although the axes may not be shown, or if they are present, the tick-marks may be spaced in a different pattern. We are going to move forward at this point to select our own WINDOW settings. WINDOW is one of the options in the GRAPH menu at the bottom of the screen.

Figure 8	We move from the GRAPH screen of Figure 7 by selecting the second option, WINDOW, from the menu. We do this by pressing the key. This brings up Figure 8, although your calculator may have different values displayed. Note that the blinking cursor is on the negative sign of the value -6.4 in Figure 8. The values in the WINDOW screen determine the WINDOW of values tht can be seen on the GRAPH screen.
Figure 9	We are going to change the settings in the WINDOW screen. We start with the xMin value. We will type to set the lowest displayed x-value, the x-value at the left edge of the screen, to be -10. This has been done for Figure 9. (NOTE: it is essential that we use the "negative" key, , and not the key when we enter this value. We want a negative value. We do not want to subtract.)
Figure 10	We press the key to move to the setting for xMax, which we change to 10, then we move down to yMin and yMax, changing them to -5 and 15, respectively. The screen should appear exactly as Figure 10. The xScl value and the yScl value in Figure 10 indicate that the graph should have a tick mark for multiple of 1 on each of the scales. That is, at 1, 2, 3, 4, 5,... and at -1, -2, -3, -4, -5,.... If the xScl value had been set to 5, then there would be a tick-mark at -10, -5, 5, and 10 on the x-axis.
Figure 11	We return to the GRAPH window by pressing the key. Now, unless there is some other problem, your calculator screen should appear exactly as is shown in Figure 11.

We continue with entering the functions into our calculator. We want to enter and graph the functions f(x)=3x-2, g(x)=3x+5, and h(x)=(-x-1)/3. We will do this one function at a time. The functions will be assigned to the successive y index = values, y1, y2, and y3.

Figure 12	To move from the GRAPH display of Figure 11 to the "Y=" display of Figure 12, we press the key to select the y(x)= menu option. The blinking cursor is positioned so that we can enter our first function.
Figure 13	We enter the function as and keys. The display should match Figure 13.
Figure 14	We had the function defined in Figure 13. Therefore, if we return to the GRAPH display by pressing the keys, to select the GRAPH menu item from the top menu, we will see the graph of the function y=3x-2 as is shown in Figure 14. Note that the ordered pairs (0,-2) and (4,10) are part of the function. Figure 14 has been augmented with red markings to identify these points. It is interesting to note that the point (-1,-5) is also on the graph, but we can not see it. The menu at the bottom of the screen is covering the bottom 8 rows of dots on the screen, including the point assigned to (-1,-5).
Figure 14a	We can remove the menu at the bottom of the screen by pressing the key. The result is shown in Figure 14a, again augmented in red with the point (-1,-5). Now that we have removed the menu, how do we get it back? Press the key to return the menu to the screen. This will restore the screen to the condition of Figure 14.

Note that the the directions for obtaining Figure 15 assume that you are coming from Figure 14, not Figure 14a. If you changed the screen to Figure 14a by clearing the menu, use the GRAPH key to restore the menu, as in Figure 14.

Figure 15	Now we want to enter the second function, y=3x+5. To do this we return to the y(x)= screen by pressing the keys, to select the y(x)= menu option from the menu. Figure 15 shows that screen. We can see the existing y1=3x-2 function, although the blinking cursor is covering the 3.
Figure 16	We press the key to move to the next line and the calculator responds by starting the line with y2=.
Figure 17	On that second line we enter . This produces the changed screen shown in Figure 17.
Figure 18	We can see the two functions defined in Figure 17 by pressing the keys. This brings up the GRAPH screen reproduced in Figure 18.

Now that we can see the graphs in Figure 18, it turns out that I do not want graphs that are so close together. We will change the problem requirements so that the second function becomes g(x)=3x+11.

Figure 19	Since we want to change the definition of the second function to g(x)=3x+11, we need to move back to the y(x)= screen by pressing the key. Again, in Figure 19, we are in the y(x)= screen with the blinking cursor covering the 3 in the first function.
Figure 20	We have moved the blinking cursor down one line by pressing the key, and then we move to the right by pressing the key 3 times. This leaves the blinking cursor covering the 5, as in Figure 20.
Figure 21	Now we can press to generate the 11 that we want.
Figure 22	We return to the GRAPH screen by pressing the keys. Figure 22 shows the graphs of the two functions. Note that the lines are parallel. This is what we expect given that they each have slope equal to 3.
Figure 23	We return to the y(x)= screen to enter the third function by pressing the key. Then we move down two lines by pressing . Finally, we insert the desired function by pressing . The result is shown in Figure 23.
Figure 24	We move to the GRAPH screen by pressing the keys. Figure 24 reproduces the graph of the three functions.
Figure 25	We can change the way the functions appear by changing the settings in the WINDOW screen. However, rather than change those settings directly, the calculator provides some special features that affect the WINDOW settings. We find these features in the ZOOM menu. We open the ZOOM menu by pressing the key. Figure 25 shows that ZOOM menu under the usual GRAPH menu. Note that there is a small arrow at the right end of the rightmost option. This arrow indicates tht there are more options in the menu and that we will have to use the key to see those additional options.
Figure 26	The ZOOM menu option that we want to explore first is ZSTD, which is the fourth option on the screen. If we press the key we will select and perform this option. The screen will be rewritten, as in Figure 26, with the new WINDOW settings. Notice in Figure 26 that the x-axis (the horizontal line) has moved to the middle of the screen. This is different from what we had in Figure 25. We can see the cause of this change if we return to the WINDOW screen so that we can review the settings there.
Figure 27	Press the keys to return to the WINDOW screen. The change from Figure 10 to Figure 27 is seen in the values assigned to yMin and yMax. In the new version, the calculator display will go from -10 at the bottom to +10 at the top. This means that the x-axis (where each y value is 0) will be in the middle of the graph.

As we look at Figure 26, we might be concerned by the fact that the graph of the last function does not appear to be perpendicular to the graphs of the first two functions. Each of the first two functions has slope=3. The last function has slope=-1/3. We may know that lines with slopes that multiply to be -1 are supposed to be perpendicular, that is, they should meet in a 90° angle. However, looking at Figure 26, those lines do not appear to be perpendicular.

The problem is in the WINDOW settings. The screen on the calculator has many more points going across the screen than it does going up and down the screen. Therefore, with a ZSTD setting, where the x-values go from -10 to 10 and the y-values go from -10 to 10, the screen is essentially stretched left and right to use the extra horizontal points on the screen. The screen is made up of lots of dots, called pixels (for picture elements), arWINDOWd in rows and columns. There are only so many vertical pixels and there are more horizontal pixels. The Y-values are spread from -10 to 10 across the vertical pixels. The X-values are spread from -10 to 10 across the horizontal pixels. Because there are more horizontal pixels, the horizontal dot-to-dot change in value is different from the vertical dot-to-dot change in value. The result of this is that lines that should be perpendicular do not appear to be perpendicular. Also, lines with a slope just slightly more than 1 will appear as lines sloping at less than 45° from the horizontal. It would be nice if we had some way to correct for this "stretching", and we do. The calculator has a ZOOM menu option that adjusts the WINDOW settings so that it compensates for the extra width in the screen.

Figure 28	For Figure 28 we re-open the ZOOM menu by pressing the key. The ZOOM option that we want, ZSQR, is not here. Therefore, we need to look at more of the menu options.
Figure 29	We press the key to view more of the ZOOM options. The ZSQR option appears in the second position. We press the key to select the ZSQR option. The result will be Figure 30.
Figure 30	Here we note that the third graph does seem to be perpendicular to the other two graphs. Furthermore, the slope of the first two graphs corresponds a bit better to our expectation for a slope of 3. Let us see what the ZSQR option did to the settings in the WINDOW screen.
Figure 31	To get to Figure 31, we press the key. Note that the yMin and yMax values remain as before in Figure 29. However, the xMin and xMax values have been changed so that we have a wider Domain. The new values mean that the dot-to-dot change in values on the x-axis corresponds to the dot-to-dot change in values on the y-axis. Therefore, with these settings, graphs will look more the way that we expect them to appear.
Figure 32	The previous Figures demonstrated the ZSTD and the ZSQR options. Now we will look at another option, the ZDECM option. We open the ZOOM menu via the key and then use the key to display the second set of options. The ZDECM option is in position 4. Pressing selects and performs that option, resulting in Figure 33.
Figure 33	The result of using the ZDECM option is the graph reproduced in Figure 33. The lines still look perpendicular, but the extent of the graph seems to have been cut down. Judging by the tick-marks, the x-values run from somewhere less than -6 to somewhere over 6, and the y-values go from just less than -3 to slightly more than 3.
Figure 34	If we press the key we return to the WINDOW screen and we can see what ZDECM has really done. Figure 34 shows us the new values with the blinking cursor covering the negative sign. The 6.3 and 3.1 values are just about what we guessed from looking at Figure 33. It just so happens that there are 63 vertical pixels and 127 horizontal pixels on the TI-83 screen. By choosing the values shown in Figure 34, the calculator is making sure that moving one pixel left, right, up, or down, corresponds to a change of exactly 0.1. Therefore, we will have pixels at points such as (2.0,3.0) and (-3.4,-1.2). That is why this setting is called ZDECM.
Figure 35	The ZOOM options that we have demonstrated so far, ZSTD, ZSQR, and ZDECM, have taken immediate effect on the graph. Now we will look at BOX, an option that requires us to press a few more buttons. We return to the ZOOM screen by pressing the key. The idea of a BOX zoom is that we will draw a box around a portion of the graph. Then we will have the calculator change the WINDOW settings so that a new GRAPH screen will be that entire box. Let us see how this works. To select BOX we merely press the key to select this option.
Figure 36	Figure 36 is the result of starting the BOX process. It looks remarkably the same as Figure 33. However, one point, one dot, one pixel, has changed. The pixel at the origin, circled in red in Figure 36, is now white. It is marking a possible location for a corner of the box that we will draw. The coordinates of that pixel are given at the bottom of the screen, namely x=0 and y=0.
Figure 37	I want to draw a box around the intersection of the first and third graphs. To do this I will move the pointer to (-0.3,0.4) by using the and keys. I have done that in Figure 37 until I have the pointer at (-0.3,0.4) as noted by the coordinates at the bottom of the screen. This point will be the upper left corner of the box that I will draw.
Figure 38	To signal that this is the location for one corner we press the key. The calcualtor responds by placing a small square at the corner point selected.
Figure 39	Now we use the key to move the pointer down to the point (-0.3,-1.1). As we do this, the calculator extends the side of the box from the corner spot that we selected.
Figure 40	We use the key to move the cursor to the right until we are at the point (1.7,-1.1), causing the calculator to draw the box from the specified corner to the pointer. In Figure 40 we can see that box. This is the box that we want to expand to the full screen size. To accept this box we press the key. That will reset the WINDOW settings and it will re-display the graphs in the new GRAPH screen shown in Figure 41.
Figure 41	Figure 41 shows the new display. It is focused onto the box that we drew in Figure 40.
Figure 42	Figure 42 represents the result of pressing the key to start drawing another box. Notice that the initial point identified in Figure 42 for a corner of a box is right in the middle of the screen, and, in fact, in the middle of the box that we had drawn in Figure 40.
Figure 43	We will use the cursor keys to re-position the cursor to the point shown in Figure 43. Notice that we have lost the nice even decimal values that we had before. This is due to the fact that the box we drew in Figure 40 did not have the 63 by 127 height to width ratio that we had established earlier. As a result, the lines no longer look to be perpendicular. We press the key to fix the point in Figure 43 as the upper right corner of the box we are going to draw.
Figure 44	In Figure 44 we have used the cursor keys to outline a box that we will expand to the entre screen.
Figure 45	We press the key to complete the BOX option. The calculator responds by drawing the screen seen in Figure 45.
Figure 46	We can look at the WINDOW setting by pressing the keys. The result appears in Figure 46. Note that the blinking cursor is covering the decimal point in the xMin value.
Figure 47	Now we return to the GRAPH screen by pressing the key. Then we will square off the screen by using the ZSQR option in the ZOOM menu. Finally, we can first hide the double menu via the key, and then recall the single menu by pressing the key. That should leave the screen as it is seen in Figure 47.
Figure 48	In Figure 48 we have pressed the key one time. A new marker, a large + sign appears. So do the coordinates of that marker. We can use that marker to move around the screen, checking the values of the points on the screen.
Figure 49	In Figure 49 we have moved the marker closer to the point of intersection of the two lines.
Figure 50	In Figure 50 we have moved the + as much onto the intersection of the two lines as we can. It is partially hidden by the lines. The coordinates suggest that we are quite close to the point (0.5,-0.5). We might notice that (0.5,-0.5) is a solution to y=3x-2 and to y=(-x-1)/3. In short, it is the point of intersection. For the given xMin, xMax, yMin, and yMax, the point (0.5,-0.5) is not one of the pixels on the screen. In a sense it is between pixels. That is why we can get close, but we can not hit it exactly on the screen in Figure 50. If we were to properly adjust the settings on the WINDOW screen we could get this to be such a point. (xMin=0.437, xMax=0.563, yMin=-0.531, yMax=-0.469 would work.)
Figure 51	We will return to the ZOOM menu to try out another option. Press , as in Figure 51. The ZOUT option allows us to determine a new center for the graph and then to increase the horizontal and vertical span around that center point. Press the key to start the ZOUT option.
Figure 52	The red circle drawn in Figure 52 is meant to help identify the blinking pointer that is at the last place we had the + pointer. We will leave that point where it is and press the key to finish the ZOUT option.
Figure 53	The ZOUT option has been performed to transform Figure 52 into Figure 53. The point has not moved, but the graph covers a larger area, as is evidenced by the appearance of the axes. Also, as expected, our point is now in the center of the screen. And, we are still in ZOUT mode. We can perform the option again merely by pressing the key, which we do to move to Figure 54.
Figure 54	Here we have moved out further. We can even see the second function over on the left side.
Figure 55	Again, we return to the ZOOM menu, first by pressing the via the key. This time we will investigate the ZIN option. We press to move to Figure 56.
Figure 56	Again, as in Figure 52, there is a marker at the intersection of the first and third functions. It is difficult to see the marker because the lines cover it. However, we can notice the white spot in the intersection. In addition, we are familiar enough with the graph to use the coordinates to locate the marker.
Figure 57	We use the cursor keys, and , to move the marker closer to the intersection of the second and third functions. Figure 57 shows that we have moved the marker. This will be the center of a new screen where we focus in on this region. We press the key to complete the ZIN option.
Figure 58	In Figure 58 we see the result of the ZIN option. The center is where we had set the marker in Figure 57. We now have a close-up view of this region. And, we are still in ZIN mode, as noted by the marker.
Figure 59	We will move even closer to the point of intersection, using the cursor keys. Figure 59 shows this new location. Press to perform another ZIN option.
Figure 60	In Figure 60 we are so close to the point of intersection that we have lost the axis from the screen. Note that we are still in ZIN mode.
Figure 61	Repeating the process one more time, we will move the marker so that it is above and to the right of the point of intersection. This is shown in Figure 61. Again, press the key to finish the ZIN option.
Figure 62	In Figure 62 we have closed in even more than before, but it is hard to tell from the screen. After all, we are looking at two straight lines and at their point of intersection. Once we are so close that we can not see other lines or the axes, the picture just does not change. We should notice the change if we moved the marker. Each time we ZIN, we decrease the change in the coordinates whenever we move the marker one pixel.
Figure 63	We can leave the ZIN mode by pressing the key. In Figure 63 we can see the two levels of the menu. We want to look at the values on the WINDOW screen. Therefore, we press to move to Figure 64.
Figure 64	Looking at Figure 64, we can see that the entire width of the graph in Figure 63 represented x-values from about -3.447073704561 to -3.25484687439, while the entire screen height represented y-values from about .758318158221 to .885334287253.
Figure 65	There is more that we want to demonstrate here. Let us return to the standard view of the GRAPH. To do this we move to the ZOOM menu via the key, then select the ZSTD option by pressing the key, and then closing the bottom menu via the key. The result is shown in Figure 65.
Figure 66	The next change that we will make will be to modify the third equation from y3=(-x-1)/3 to y3=(-x+13)/3. To do this, first move to the y(x)= screen by pressing the key. Then, we use the cursor keys to move the blinking cursor down to the third line and then across to the subtraction sign, as shown in Figure 66.
Figure 67	In Figure 66 we positioned the cursor on top of the subtraction sign. Now we replace that subtraction with a plus sign by pressing the key. Then we move to the right two places by using the key twice. This leaves us with the blinking cursor on top of the right parenthesis, as is shown in Figure 67.
Figure 68	We want to insert a character here. We move the calculator into INSERT mode by pressing . The blinking block cursor changes to a blinking underline cursor to indicate that we are in INSERT mode. Figure 68 has been augmented with a red parenthesis to remind us that the parenthesis character is still there.
Figure 69	The character that we want to insert is the number 3, so we press the key. We have done this in Figure 69. Since we are in INSERT mode, the 3 is inserted in the text. The blinking underline cursor moves to the right, along with the rest of the line. Again, the figure has been augmented with the red parenthesis.
Figure 70	We exit INSERT mode and return to the GRAPH screen by pressing the keys. The new plot shows the change in the third equation.
Figure 71	We have looked at four of the five menu items in the main menu shown at the bottom of the screen. Let us check out the remaining menu item. If we press the key, the calculator moves into TRACE mode, shown in Figure 71. The display has been changed by placing the number 1 at the top right corner of the screen. In addition, a blinking marker is placed on the first equation, in this case at the point (0,-2). And the coordinates of the location of the blinking marker have been displayed at the bottom of the screen. In TRACE mode, we use the left and right cursor keys to move the blinking marker along the equation identified by the number at the top right of the screen. We use the up and down cursor keys to make the blinking marker jump from one equation to the next.
Figure 72	For Figure 72 we have used just the key to move the blinking cursor along the first equation. Note the changing coordinates as you do this. We stop, in Figure 72, when x=1.4285714286.
Figure 73	To move from Figure 72 to Figure 73 press the key. This moves the trace from the first equation, y1=3x-2, to the second equation, y2=3x+11. Notice that the number 2 is now displayed at the top right of the screen. But, where is the blinking cursor? If we look at the coordinates at the bottom of the screen, we see that the x-value has not changed from what it was in Figure 73. However, the corresponding y-value for the second equation is y=15.285714286, a point that is off of the screen. Therefore, we are tracing the second equation, but our marker is off the screen at this point.
Figure 74	We can press the key to jump the trace to the third equation. The x-value has not changed, but the corresponding y-value is now on the screen. Therefore, we can see the blinking marker sitting on the third line.
Figure 75	We will use the key to move that blinking marker to the intersection of the two lines. We can see, in Figure 75, that we are close to the intersection, that we are tracing the third equation, y3=(-x+13)/3, that the x-value is 1.90447619048 and the y-value is 3.6984126984.
Figure 76	For Figure 76 we have pressed the key so that we are again tracing the y1=3x-2 equation. The x-value remains unchanged from Figure 75. However, there is a new y-value, namely, 3.3.7142857144. The two lines are not at the same y-value for the given x-value. In other words, we are not really at the point of intersection. Let us zoom in on the picture.
Figure 77	For Figure 77 we have shifted to the ZOOM menu by pressing the key to stop the TRACE and remove the coordinates from the display, then the key to restore the menus, and then to open the ZOOM menu. To leave Figure 77 and move to Figure 78 we press the key to start the ZIN option.
Figure 78	In Figure 78 we are in the middle of the ZIN option. Although it is hard to recognize on the printed screen (because it is not blinking here) the marker is right where we left it. It does show as a single pixel white spot near the point of intersection.
Figure 79	We move from Figure 78 to Figure 79 by pressing the key to perform the ZIN option. Notice that we can see that the marker is slightly separated from the lines.
Figure 80	For Figure 80 we have returned to the ZOOM menu and selected the BOX option. Then we have located the corner of the box and we have pressed the key to fix that location.
Figure 81	This picture shows the marker as moved to the opposite corner of the box.
Figure 82	Having drawn the box in the previous screens, we press the key to complete the BOX option. The result is shown in Figure 82.
Figure 83	At this point we can press the key three times. This will make the + marker appear and move up from the center of the screen.
Figure 84	In Figure 84 we have moved the + marker so that is approximately on top of the intersection of the two lines. Judging from the coordinates at the bottom of the screen, the lines cross at or near (1.9,3.7). Let us manufacture a screen that uses decimal values and that includes this point.
Figure 85	We can return to the WINDOW screen by pressing . Figure 85 gives the current values of the WINDOW settings. To get decimal values we want to have 127 points going across and 63 points going up and down.
Figure 86	If we want the screen to have (1.9,3.7) as the center, then we can set the xMin to 1.9-0.0063, xMax to 1.9+0.0063, yMin to 3.7-0.0031 and yMax to 3.7+0.0031. We can do the arthmetic in our heads and enter the correct values, or we can actually enter the expressions (e.g., 1.9-0.0063) and the calculator will simplify the values. Figure 86 shows the status of the screen after all values have been entered.
Figure 87	By pressing the key we leave the WINDOW screen and move to the GRAPH screen shown in Figure 87.
Figure 88	Pressing the key moves the calculator into TRACE mode. Figure 88 shows that we are at the point (1.9,3.7) and that we are tracing the equation number 1. We are at that point because it is the center of the newly created screen.
Figure 89	We can shift the TRACE focus from equation 1 to equation 3 by pressing the key 2 times. Figure 89 shows that we are still at the same point, but now we are on equation 3. This must be the point of intersection since it is a point on both lines.
Figure 90	Figure 90 is the result of moving the TRACE marker to the right. At each step of the move the coordinates change. It is important to note that the coordinates are those of the real line, not of the selected pixels. After all, a close inspection of the line for equation 3 reveals that, in Figure 90, it is made up of groups of three horizontally adjacent pixels. Remember that the equation has a slope of 1/3. If we have created a screen with equal horizontal and vertical spacing, then it should take three horizontal pixels for each vertical one. However, as the TRACE marker moves across the three pixels, the y-coordinate will show the appropriate changing value.
Figure 91	Pressing from Figure 90 will cause the TRACE marker to move to the first equation, as is shown on Figure 91.

That is all for this page. Hopefully, it has given you a chance to do some graphing and to learn to use the various keys and options on the calculator.

There are many other graphing features on this calculator. Their explanation and illustration will have to wait for another page.