Graphing on the TI-83

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 brings up a display of all of the functions that are currently defined for graphing on the calculator. Each separate function starts with Ysubscript= 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.
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. 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.

Figure 8	Figure 8 shows a typical empty graph. Note that the axes are displayed and that there are numerous small "tick-marks" on the axes. Figure 8 has been augmented in red to point out those tick-marks. Note that it is entirely possible that your calculator does not show the axes. There is a possible reason for this, namely that the WINDOW being displayed does not contain the axes. The correction for this is covered in Figures 9 through 12.
Figure 9	As it is shown, Figure 8 does not indicate the value associated with the tick-marks. We want to see the limits of the x and y values on the screen. To do this we move to the "WINDOW" screen by pressing the key. Figure 9 shows the values that were in effect for the calculator used here. Of course, your calculator may have different values.
Figure 10	We will agree on the limits that we want to use for our display. In particular, we want the left edge of the display to correspond to a x value of -10. To do this we press the keys . (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.) This leaves the screen as seen in Figure 10.
Figure 11	Figure 11 represents a "WINDOW" screen with new values. The Xmin value was set in Figure 10. We use the key to move down to the Xmax value, which we set to 10. Then the keys to move to the Ymin field. We set that to negative five (-5) and go down to the Ymax field which we set to 15. This is the condition shown in Figure 11. The Xscl value and the Yscl value in Figure 11 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. The Xres value indicates that the graph should be drawn by plotting a Y value for every X value along the X-axis. If the Xres value had been set to 2, then the graph will be drawn by plotting Y value for every second X value along the X-axis.
Figure 12	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 12.

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_subscript values, Y₁, Y₂, and Y₃.

Figure 13	To move from the GRAPH display of Figure 12 to the "Y=" display of Figure 13, we press the key. The blinking cursor is positioned so that we can enter our first function.
Figure 14	We enter the function as and keys. The display should match Figure 14.
Figure 15	We had the function defined in Figure 14. Therefore, if we return to the GRAPH display by pressing the key we will see the graph of the function Y=3X-2 as is shown in Figure 15. Note that the ordered pairs (0,-2) and (-1,-5) and (4,10) are part of the function. Figure 15 has been augmented with red markings to identify these points.
Figure 16	Now we want to enter the second function, Y=3X+5. To do this we return to the Y= screen by pressing the key. Figure 16 shows that screen. We can see the existing Y1=3X-2 function, although the blinking cursor is covering the 3.
Figure 17	We press the key to move to the Y2= line. There 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 key. 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= screen by pressing the key. Again, in Figure 19, we are in the Y= 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 key. 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= 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 key. 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 screen. We open the ZOOM screen by pressing the key. Figure 25 shows that ZOOM screen with the first choice, 1:ZBox, highlighted.
Figure 26	The ZOOM screen option that we want to explore first is 6:ZStandard. Figure 26 shows that we have moved down to the 6:ZStandard line by pressing the key 5 times. Once the 6:ZStandard line is highlighted, we can select it by pressing the key. (Alternatively, we can select option 6, without even highlighting it, by pressing the key. This is a faster method. It selects and executes a line in one push of a key. I will not employ that method here because it is harder to follow with pictures.)
Figure 27	Having selected the ZStandard option, the calculator returns to the GRAPH screen and re-displays the graph. Notice in Figure 27 that the x-axis (the horizontal line) has moved to the middle of the screen. This is different from what we had in Figure 24. We can see the cause of this change if we return to the WINDOW screen so that we can review the settings there.
Figure 28	Press the key to return to the WINDOW screen. The change from Figure 11 to Figure 28 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 27, 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 27, 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 ZStandard 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), arranged 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 option that adjusts the WINDOW settings so that it compensates for the extra width in the screen.

Figure 29	For Figure 29 we have moved back to the ZOOM screen by pressing the key. The option that we want is 5:ZSquare and we have used to key to move the highlight down to that option. Now, to move to Figure 30, pess the key
Figure 30	Once we have selected the ZSquare option, the calculator re-displays the graph, as shown in 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 ZSquare option did to the settings in the WINDOW screen.
Figure 31	We move back to the window screen by pressing the key. The result is given in Figure 31. Note that the Ymin and Ymax values remain as before in Figure 28. 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 ZStandard and the ZSquare options. Now we will look at another option, the ZDecimal option. We return to the ZOOM screen via the key. Then we move the highlight down to the 4:ZDecimal option with the key.
Figure 33	We select the ZDecimal option in Figure 32 by pressing the key. The result 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 -4 to somewhere over 4, and the y-values go from just less than -3 to slightly more than 3.
Figure 33a	If we press the key we return to the WINDOW screen and we can see what ZDecimal has really done. Figure 33a shows us the new values with the blinking cursor covering the negative sign. The 4.7 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 95 horizontal pixels on the TI-83 screen. By choosing the values shown in Figure 33a, 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 ZDecimal.
Figure 34	The ZOOM options that we have demonstrated so far, ZStandard, ZSquare, and ZDecimal, have taken immediate effect on the graph. Now we will look at ZBox, 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 ZBox 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. Since 1:ZBox is already highlighted, merely press the key to select this option.
Figure 35	Figure 35 is the result of starting the ZBox 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 35, 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 36	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 36 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 37	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 38	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 39	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 39 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 40.
Figure 40	Figure 40 shows the new display. It is focused onto the box that we drew in Figure 39. In addition, we are still in ZBox mode. Note the pointer at position (0.7,-0.35). This is the point precisely in the center of the box that we drew in Figure 39, and it is the point in the center of Figure 40.
Figure 41	Figure 41 represents the result of moving the pointer one pixel to the left and one pixel down from where it was in Figure 40. We did that by pressing the key one time, and then presing the key one time. Notice that we no longer have nice decimal values for the pixels. That is because we have ruined the required ratio of length to width for the screen.
Figure 42	We can continue to box in the intersection of the two lines. We press the key to mark a corner of our next box. Figure 42 shows that corner has been set.
Figure 43	We use the cursor keys to move to the opposite corner position. The calculator draws the box around the area we wish to expand to a full screen. Figure 43 shows that box. We press the key to move to Figure 44.
Figure 44	The box that we created in Figure 43 has been expanded to the full screen of Figure 44. Again our ratios are off so lines that should be (and are) perpendicular do not appear to be so on the screen. We are still in ZBox mode as noted by the marker and the coordinate display.
Figure 45	Pressing the key will take us to the WINDOW screen so that we can see just how the ZBox operations that we have performed have changed the settings. Note in Figure 45 that the limits on x and y force us to focus in on a small area of the coordinate plane.
Figure 46	We can return to the ZOOM screen, by pressing , and we can use one of the earlier options, ZSquare, to adjust the x and y limits to make our graph have equal vertical and horizontal movements. As before, we move to highlight the option we want and then we press to accept that option.
Figure 47	Figure 47 shows a new graph, but this time adjusted for equal horizontal and vertical movement. Note tht the lines appear to be perpendicular, again.
Figure 48	Figure 48 shows the result of pressing the key. The calculator shifts into a mode where it displays a + as a marker, along with the coordinates of the pixel at the center of the +.
Figure 49	Figure 49 shows the screen as we have moved the + to the right and down, moving toward 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=0.453, Xmax=0.547, Ymin=-0.531, Ymax=-0.469 would work.)
Figure 51	We will return to the ZOOM screen to try out another option. Press and then use the key two times to move the 3:ZOut option, 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 a portion of the second function over on the left side.
Figure 55	Again, we return to the ZOOM screen via the key. This time we will investigate the 2:ZIn option. We move the highlight down to the second line, and then 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, and then press to perform another ZIn.
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	Even without leaving the ZIn mode, we can move the marker to the point of intersection to see if we can get a good approximation to the coordinates of that point. From Figure 63, where the marker has been moved to the approximate point of intersection, we might expect x=-3.4 and y=0.8.
Figure 64	In Figure 64 we return to the WINDOW screen by pressing the key. We can see that the entire width of the graph in Figure 63 represented x-values from about -3.445031 to -3.312052, while the entire screen height represented y-values from about .76247791 to .8501728.
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 screen via the key, and then move the highlight to the 6:ZStandard option.
Figure 66	We move from Figure 65 to Figure 66 by pressing the key to accept the ZStandard option. This re-displays the graph. Notice that we have lost the sense of equal horizontal and vertical scales by doing this. The lines no longer appear perpendicular.
Figure 67	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= screen by pressing the key. Then, we use the cursor keys to move the blinking cursor onto the subtraction sign, as shown in Figure 67.
Figure 68	In Figure 67 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 68.
Figure 69	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 69 has been augmented with a red parenthesis to remind us that the parenthesis character is still there.
Figure 70	The character that we want to insert is the number 3, so we press the key. We have done this in Figure 70. 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 71	We exit INSERT mode and return to the GRAPH screen by pressing the key. The new plot shows the change in the third equation.
Figure 72	We have looked at four of the five keys at the top of the keypad. Let us check out the remaining key. If we press the key, the calculator moves into TRACE mode. The display has been changed by placing a copy of the first equation at the top 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 displayed at the top. We use the up and down cursor keys to make the blinking marker jump from one equation to the next.
Figure 73	For Figure 73 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 73, when x=1.4893617.
Figure 74	To move from Figure 73 to Figure 74 press the key. This moves the trace from the first equation, Y1=3X-2, to the second equation, Y2=3X+11. Notice that it is the second equation that is now displayed at the top left 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=15.468085, 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 75	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 76	We will use the key to move that blinking marker to the intersection of the two lines. We can see, in Figure 76, that we are close to the intersection, that we are tracing Y3=(-X+13)/3, that the x-value is 1.918936 and the y-value is 3.6950355.
Figure 77	For Figure 77 we have pressed the key so that we are again tracing the Y1=3X-2 equation. The x-value remains unchanged from Figure 76. However, there is a new y-value, namely, 3.7446809. 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 78	For Figure 78 we have shifted to the ZOOM screen by pressing the key. Then we moved down to the ZIn option. To leave Figure 78 and move to Figure 79 we press the key.
Figure 79	In Figure 79 we are in the middle of the ZIn option. We need to place the blinking cursor at the point we want to be the center of the new screen. Once that is done, we press to perform the ZIn option.
Figure 80	Here, in Figure 80, we have zoomed in to the area around the point of intersection. We can see that we are not at the point of intersection, but we are close. Let us zoom in further, but this time we will use the ZBox option.
Figure 81	We return to the ZOOM screen via the key. ZBox is the default selection. We need only press the key to start the process, and move to Figure 82.
Figure 82	We are in the ZBox process. We need to place the marker at a corner of the box that we want to draw. We will use the cursor keys to move the marker to the position shown in Figure 82. Then we press the key to set that corner.
Figure 83	In Figure 83 we have used the cursor keys to move the cursor to the opposite corner of the box. The calculator expands the box as we move the cursor.
Figure 84	Pressing the key accepts this point as the opposite corner and uses that region to define a new area for the next GRAPH screen, shown in Figure 84. Note that the blinking cursor is right in the middle of the graph, covered a bit by the line.
Figure 85	We can use the cursor keys to move the blinking cursor to the intersection of the two lines, as has been done in Figure 85. This gives us a good approximation to the true value of the coordinates of the point of intersection, somewhere close to (1.9,3.7).
Figure 86	In Figure 86 we press the key to move back to the WINDOW screen to check out the current settings.
Figure 87	Rather than use the ZOOM options, we can perform our own ZOOM by changing the settings on the WINDOW screen. If we want the point (1,9,3.7) to be at the center of the screen, then we need to change the WINDOW settings so that those values are exactly between the low and high values for both x and y. In addition, we recall that the screen is 63 pixels high by 95 pixels wide. Therefore, we can chose our x-limits as 1.9±0.0047 and our y-limits as 3.7±0.0031. Those values will not only put the point (1,9,3.7) in the middle, they will also give us a display with equal horizontal and vertical spacing, and they will give us pixels with coordinates that change by 0.0001 in each direction. Those values are shown in Figure 87.
Figure 88	We press the key to move from Figure 87 to Figure 88. We observe in Figure 89 that we have been successful in placing the intersection in the middle of the graph.
Figure 89	For FIgure 89 we have moved into TRACE mode by pressing the key. Note that the first equation is displayed at the top left. Also, note that the marker is at (1.9,3.7).
Figure 90	If we use the key to shift the trace to the third function we see Figure 90. Again, the marker is at the same point. We have found the exact point of intersection of the two lines.

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.