Looking at Pixels on the TI-83

The TI-83 has a display with 5,985 little squares on it. Those squares are arranged in 95 columns and 63 rows. The individual squares are called pixels, a name arising from television as a shortened version of picture element, i.e., the little pieces that make up a picture. As a reference, note that a normal computer monitor has well over 480,000 pixels. Whenever the TI-83 needs to plot or graph a point, (x,y), the calculator determines the pixel that represents that value and the TI-83 sets that pixel to appear black. In order to determine just which pixel it should use, the calculator depends upon values that signify the smallest and the largest values that can be shown horizontally and vertically on the screen. The user of the TI-83 can set these minimum and maximum values. Once the values have been set, the TI-83 can determine the values associated with each pixel. This page illustrates this material by plotting specific points on the graph and examining the resulting image.

Figure 1	We start this sequence on the Y= screen by pressing the key. We note in Figure 1 that the calculator used here has a function defined for Y1. For the purpose of this page we do not want that function defined at this time. Therefore, we will clear the function by pressing the key. This will produce Figure 2.
Figure 2	Figure 2 has no functions defined for graphing. If need be, we could have cleared other functions by moving the cursor to those functions and pressing the CLEAR key. The important point here is that we have cleared all graphing functions.
Figure 3	We move to Figure 3 by pressing the key. That will bring up the ZOOM menu shown here. We can select any of the options, but the one that we will use is option 4: ZDecimal We select that option by pressing the key. This will set certain default values for the minimum and maximum values that can be displayed on the screen, and it will move us to the graph screen shown in Figure 4.
Figure 4	The calculator used here did not have any plots of points defined, and it did not have any graphing functions defined (we cleared those in Figures 1 and 2). Therefore, the graph has nothing on it but the axes. We can tell a good deal from the graph of the axes. Figure 4A is a reproduction of Figure 4, but with certain features identified.
Figure 4A	The ZDecimal option that we used to generate Figure 4 sets the parameters for the screen so that the middle of each pixel has coordinates that are exact decimal values to one decimal place to the right of the decimal point. That is, these points all have coordinates that are in even tenths. In Figure 4A we have identified the "tick marks" that help us divide up the X and Y scales. In addition, we have identified the particular pixels that represent certain ordered pairs. We can see the actual settings for the minimum and maximum values, and the frequency for the "tick marks" by moving to the WINDOW screen. We do this by pressing the key. This will move us to Figure 5.
Figure 5	Figure 5 was captured from the calculator screen as the blinking cursor covered the minus sign before the Xmin value of -4.7. In Figure 5 we can see the values that have been assigned to Xmin, Xmax, Ymin, and Ymax. If we remember that there are 95 columns of pixels, then we can see that we can assign the Xmin value, -4.7, to the leftmost column, -4.6 to the next column to the right, -4.5 to the next column, and so on. Eventually, we will assign the middle column to 0 and then the next column to 0.1, and so on until we get to the 95^th column, which we will assign to 4.7, which is the Xmax value. Similarly, there are 63 rows of pixels. We can assign the bottom row to -3.1, the row above it to -3.0, and so on, including the middle row being assigned to 0 and the 63^rd row being assigned to 3.1. Note that there is another value on the screen, namely Xres. We will discuss that value in Figure 23.

The text above was careful to imply that values are associated with pixels. Once the Xmin, Xmax, Ymin, and Ymax values have been set, we can determine the boundaries for each pixel. The figure at the right is meant to represent a small portion, 9 pixels, from the graph above. Each pixel has a small dot in its center and the coordinates of that dot are given inside each cell. We note that there is much more to each pixel than the labeled point in the center. On the outside of the figure we have the values that must be at the boundary of each pixel. Thus, the very center pixel, the one with the point (-2.0,1.4) must represent all of the X-values between -2.05 and -1.95 paired with Y-values between 1.35 and 1.45. If the calculator is trying to graph or plot a point that falls into the ranges just given, then it is the center pixel in the figure that will get turned black. Thus, each of the points

(-2,1.4)
(-2.03,1.42)
(-2.04478,1.35001)
(-1.95123,1.4444)
((2-)/0.5575545, /2.21863) will be associated with that same center pixel.

Note that there is a small problem here. The point

(-2.0,1.45) falls right on the border between two pixels. Which one is chosen? The TI-83 will choose the upper-center pixel for that point. That is, the horizontal boundaries go to the cell above them. This leaves the question of the vertical boundaries. Which pixel will get the point (-2.05,1.4)? The answer is that the vertical boundaries go with the cell to the right. Thus, the point (-2.05,1.4) will be assigned to the center pixel.

We will use the TI-83 to generate a plot of six different points in order to illustrate some of this.

Figure 6 To generate a plot we need to enter some values into lists. The TI-83 has six standard lists. We will use L1 and L2. To do this, we want to make sure that those lists are in the list editor. We can do this first step from the STAT menu. We press the key to open the STAT menu as seen in Figure 6.
From this menu, we need to choose option 5: SetUpEditor. We do this by pressing the key to paste SetUpEditor onto the main screen.

Figure 7 Figure 7 shows the main screen after we have pasted SetUpEditor and after we have pressed the key to have the calculator perform that command. The only confirmation that we get is the word Done on the right of the screen.

Figure 8 The next step is just a convenience for us. We will clear all of the lists on the calculator. This will save us from having to worry about values that were previously stored. We will clear all of the lists by using the command ClrAllLists on the MEM menu. To do this we press to open the menu, shown in Figure 8. The option we want is #4. We press the to select this option.

Figure 9 Figure 9 shows the result of making our selection and of pressing the key to perform the command. Again, the calculator responds with the word Done.

Figure 10 We return to the STAT menu, via the key, so that we can start to edit the lists L1 and L2. We press the key to select the highlighted option, 1:Edit. This will move us to Figure 11.

Figure 11 We are ready to start entering values into the lists. The highlight is on the spot where we can enter the first value for L1. The points that we want to enter are: (-4.66,2.9)
(-4.64,2.5)
(-4.56,2.0)
(-4.54,1.5)
(-4.7,1.0)
(-4.71,0.5) The X-coordinates need to be placed into the L1 list, and the Y-coordinates need to be placed into the L2 list.

Figure 12 We start by entering the -4.66, pressing the keys, and then . The result is shown in Figure 12. Note that we are ready for the next X-coordinate.

Figure 13 Figure 13 shows the list after we have entered each of the six X-coordinates. The calculator is ready to receive a seventh, but we are done withour values. Therefore, we press the key to move to the L2 list.

Figure 14 We have moved to the L2 list and the calculator is ready to accept the Y-coordinates.

Figure 15 In Figure 15 we have completed entering all of the coordinates. We are done with the data entry process. We can QUIT this by pressing . This returns us to the main screen, shown in Figure 16, just as we left it in Figure 9.

Figure 16 At this point we have entered the values in the two lists, but we have not told the calculator that we want it to plot those values. We can start this process by moving to the STAT PLOT screen.

Figure 17 We get to the STAT PLOT screen by pressing . Figure 17 shows the result. This screen shows four options. The first option, if selected, will allow us to change the settings associated with Plot1. As a convenience to us, the calculator shows, on the STAT PLOT screen, the settings that are in effect for PLOT1. First, Plot1 is set "Off". This means that it will not be plotted. Second, the plot is to be a scatter plot. Third, Plot1 is going to use L1 and L2 . And, fourth, Plot1 will use the square symbol, , to mark each of the points to be plotted. The one setting that we need to change is the first, from Off to On. The STAT PLOT screen does not allow us to make this change. Instead, we need to select option 1 to move us to a new screen, dedicated to the Plot1 settings. We can select option 1 by pressing the key.

Figure 18 Figure 18 shows the screen that allows us to change the settings for Plot1. When Figure 18 was captured from a calculator, the blinking cursor was covering the "On" option. The "Off" value is highlighted because it is the current setting. Since the cursor is at the "On" option, we can change that setting by pressing the key. THis will change the display to that of Figure 19.

Figure 19 Note in Figure 19 that the blinking cursor is still at the "On" value, but the "On" value is also highlighted. We have turned "On" Plot1. We can see the effect of this change by pressing the key to display the graph.

Figure 20 Here we have the plot of the points in L1 and L2. We need to examine this plot with respect to the values that we know are being plotted and remembering the WINDOW settings that are in effect.
First, note that although we defined six points in L1 and L2, there are only five points that are plotted. The last point, (-4.71,0.5), has not been plotted. If we recall the true limits to each pixel, we can calculate that the leftmost column of pixels are associated with the numbers -4.75 through -4.65. Our sixth point, (-4.71,0.5), falls within this range. However, for the pixels at the edge of the screen, the calculator uses the Xmin, Xmax, Ymin, and Ymax values as the limit to what it will plot, even if the desired value is just slightly outside of the given range.

Figure 20A (Augmented) Figure 20 has been repeated in an augmented form in Figure 20A. Here we have given the coordinates of the five points that were plotted. We note that the topmost point has coordinates (-4.66,2.9) and that point should be in the leftmost column of the screen, the one associated with the range of X-values -4.75 through -4.65. Even though we are just into that range, it is indeed the leftmost column that is identified, by having the small square, , placed around it. Indeed, because this is the leftmost column, the left third of the square can not be shown.
The next two points have coordinates
(-4.64,2.5) and (-4.56,2). Both fall into the range of values associated with the second column of pixels, namely, values from -4.65 through -4.55. Therefore, the pixels that are associated with these points are both in that same second column of pixels.

Figure 20A (Repeated) Again, in Figure 20A, the fourth point has coordinates (-4.54,1.5). The X-coordinate, -4.54, is just less than the right boundary value for the second column of pixels, -4.55. Therefore, the fourth point is plotted in the next, the third, column of pixels.
The last point to be plotted has the X-coordinate -4.7. This is the value of Xmin. It is the lowest X-value that can be plotted on this graph.

Figure 21 Until now we have plotted points. But what if we want to graph a function such as Y=4X+1? First we need to enter the function. We press to move to the Y= screen shown in Figure 21.

Figure 22 In Figure 22 we have defined the function by pressing the keys

Figure 23 We return to the GRAPH screen by pressing . Note that we did not turn off the plot, so our points are still plotted. However, in Figure 23 we also have the graph of the function. A close examination of the graph of the function shows that it is made up of a number of small vertical segments. Why?
To generate the graph, the TI-83 tries different values of X and it generates the corresponding values for Y. What values does it use? It starts at the minimum X-value, Xmin, and it tries that value. Then it finds the next value by moving to the center X-value in a pixel Xres steps to the right. Xres was defined in the WINDOW settings that we saw back in Figure 5. Thus, we start X at the minimum value, -4.7, and get a value for Y. If that point can be plotted, then the calculator graphs it. It then moves Xres pixels to the right. Since Xres is 1 for this graph, the calcualtor selects the center value in the second column, namely -4.6. Again, the calculator uses that X-value to determine a Y-value, and the resulting point is plotted if it is on the screen. This process is repeated until we have moved across all 95 columns of pixels.
Thus, for the function Y=4X+1, the X-value 0.3 should produce the Y-value 2.2, 0.4 should produce 2.6, and 0.5 should produce 3.0. These points, (0.3,2.2), (0.4,2.6), and (0.5,3.0), would be graphed in adjacent columns of pixels, but not adjacent rows. You can identify each of the pixels in the following, magnified and enhanced image of a portion of Figure 23.

Figure 24 The explanation for the extra pixels in the graph of the function Y=4X+1 can be found in the MODE screen. Pressing the key should display a screen similar to that in Figure 24. Note that the fifth row of options has the value CONNECTED highlighted. That setting tells the calculator to fill in extra pixels in cases such as the Y=4X+1 shown above, where a pixel in one column is not touching (on the corner or on the side) the calculated pixel in the next column.

Figure 25 The alternate to CONNECTED is DOTS. We can move to select DOTS by pressing the key four times to move the cursor to that row of options, and then pressing the key to highlight the DOTS option. Figure 25 shows such a condition, with the blinking cursor covering the DOTS option.

Figure 26 To actually make the change we need to press the key. Figure 26 shows the option changed to DOTS.

Figure 27 We can return to the GRAPH screen by pressing the key. The new graph is shown in Figure 27. Note that the connecting pixels have not been turned on. A close examination will show that the pixels associated with the points we selected earlier, (0.3,2.2), (0.4,2.6), and (0.5,3.0), are exactly the pixels that have been turned black. To facilitate your examination, Figure 27 is reproduced below, but 3 times as wide and 3 times as tall.

Here is a reproduction of Figure 27, but magnified.

Figure 6	To generate a plot we need to enter some values into lists. The TI-83 has six standard lists. We will use L1 and L2. To do this, we want to make sure that those lists are in the list editor. We can do this first step from the STAT menu. We press the key to open the STAT menu as seen in Figure 6. From this menu, we need to choose option 5: SetUpEditor. We do this by pressing the key to paste SetUpEditor onto the main screen.
Figure 7	Figure 7 shows the main screen after we have pasted SetUpEditor and after we have pressed the key to have the calculator perform that command. The only confirmation that we get is the word Done on the right of the screen.
Figure 8	The next step is just a convenience for us. We will clear all of the lists on the calculator. This will save us from having to worry about values that were previously stored. We will clear all of the lists by using the command ClrAllLists on the MEM menu. To do this we press to open the menu, shown in Figure 8. The option we want is #4. We press the to select this option.
Figure 9	Figure 9 shows the result of making our selection and of pressing the key to perform the command. Again, the calculator responds with the word Done.
Figure 10	We return to the STAT menu, via the key, so that we can start to edit the lists L1 and L2. We press the key to select the highlighted option, 1:Edit. This will move us to Figure 11.
Figure 11	We are ready to start entering values into the lists. The highlight is on the spot where we can enter the first value for L1. The points that we want to enter are: (-4.66,2.9) (-4.64,2.5) (-4.56,2.0) (-4.54,1.5) (-4.7,1.0) (-4.71,0.5) The X-coordinates need to be placed into the L1 list, and the Y-coordinates need to be placed into the L2 list.
Figure 12	We start by entering the -4.66, pressing the keys, and then . The result is shown in Figure 12. Note that we are ready for the next X-coordinate.
Figure 13	Figure 13 shows the list after we have entered each of the six X-coordinates. The calculator is ready to receive a seventh, but we are done withour values. Therefore, we press the key to move to the L2 list.
Figure 14	We have moved to the L2 list and the calculator is ready to accept the Y-coordinates.
Figure 15	In Figure 15 we have completed entering all of the coordinates. We are done with the data entry process. We can QUIT this by pressing . This returns us to the main screen, shown in Figure 16, just as we left it in Figure 9.
Figure 16	At this point we have entered the values in the two lists, but we have not told the calculator that we want it to plot those values. We can start this process by moving to the STAT PLOT screen.
Figure 17	We get to the STAT PLOT screen by pressing . Figure 17 shows the result. This screen shows four options. The first option, if selected, will allow us to change the settings associated with Plot1. As a convenience to us, the calculator shows, on the STAT PLOT screen, the settings that are in effect for PLOT1. First, Plot1 is set "Off". This means that it will not be plotted. Second, the plot is to be a scatter plot. Third, Plot1 is going to use L1 and L2 . And, fourth, Plot1 will use the square symbol, , to mark each of the points to be plotted. The one setting that we need to change is the first, from Off to On. The STAT PLOT screen does not allow us to make this change. Instead, we need to select option 1 to move us to a new screen, dedicated to the Plot1 settings. We can select option 1 by pressing the key.
Figure 18	Figure 18 shows the screen that allows us to change the settings for Plot1. When Figure 18 was captured from a calculator, the blinking cursor was covering the "On" option. The "Off" value is highlighted because it is the current setting. Since the cursor is at the "On" option, we can change that setting by pressing the key. THis will change the display to that of Figure 19.
Figure 19	Note in Figure 19 that the blinking cursor is still at the "On" value, but the "On" value is also highlighted. We have turned "On" Plot1. We can see the effect of this change by pressing the key to display the graph.
Figure 20	Here we have the plot of the points in L1 and L2. We need to examine this plot with respect to the values that we know are being plotted and remembering the WINDOW settings that are in effect. First, note that although we defined six points in L1 and L2, there are only five points that are plotted. The last point, (-4.71,0.5), has not been plotted. If we recall the true limits to each pixel, we can calculate that the leftmost column of pixels are associated with the numbers -4.75 through -4.65. Our sixth point, (-4.71,0.5), falls within this range. However, for the pixels at the edge of the screen, the calculator uses the Xmin, Xmax, Ymin, and Ymax values as the limit to what it will plot, even if the desired value is just slightly outside of the given range.
Figure 20A (Augmented)	Figure 20 has been repeated in an augmented form in Figure 20A. Here we have given the coordinates of the five points that were plotted. We note that the topmost point has coordinates (-4.66,2.9) and that point should be in the leftmost column of the screen, the one associated with the range of X-values -4.75 through -4.65. Even though we are just into that range, it is indeed the leftmost column that is identified, by having the small square, , placed around it. Indeed, because this is the leftmost column, the left third of the square can not be shown. The next two points have coordinates (-4.64,2.5) and (-4.56,2). Both fall into the range of values associated with the second column of pixels, namely, values from -4.65 through -4.55. Therefore, the pixels that are associated with these points are both in that same second column of pixels.
Figure 20A (Repeated)	Again, in Figure 20A, the fourth point has coordinates (-4.54,1.5). The X-coordinate, -4.54, is just less than the right boundary value for the second column of pixels, -4.55. Therefore, the fourth point is plotted in the next, the third, column of pixels. The last point to be plotted has the X-coordinate -4.7. This is the value of Xmin. It is the lowest X-value that can be plotted on this graph.
Figure 21	Until now we have plotted points. But what if we want to graph a function such as Y=4X+1? First we need to enter the function. We press to move to the Y= screen shown in Figure 21.
Figure 22	In Figure 22 we have defined the function by pressing the keys
Figure 23	We return to the GRAPH screen by pressing . Note that we did not turn off the plot, so our points are still plotted. However, in Figure 23 we also have the graph of the function. A close examination of the graph of the function shows that it is made up of a number of small vertical segments. Why? To generate the graph, the TI-83 tries different values of X and it generates the corresponding values for Y. What values does it use? It starts at the minimum X-value, Xmin, and it tries that value. Then it finds the next value by moving to the center X-value in a pixel Xres steps to the right. Xres was defined in the WINDOW settings that we saw back in Figure 5. Thus, we start X at the minimum value, -4.7, and get a value for Y. If that point can be plotted, then the calculator graphs it. It then moves Xres pixels to the right. Since Xres is 1 for this graph, the calcualtor selects the center value in the second column, namely -4.6. Again, the calculator uses that X-value to determine a Y-value, and the resulting point is plotted if it is on the screen. This process is repeated until we have moved across all 95 columns of pixels. Thus, for the function Y=4X+1, the X-value 0.3 should produce the Y-value 2.2, 0.4 should produce 2.6, and 0.5 should produce 3.0. These points, (0.3,2.2), (0.4,2.6), and (0.5,3.0), would be graphed in adjacent columns of pixels, but not adjacent rows. You can identify each of the pixels in the following, magnified and enhanced image of a portion of Figure 23.

Figure 24	The explanation for the extra pixels in the graph of the function Y=4X+1 can be found in the MODE screen. Pressing the key should display a screen similar to that in Figure 24. Note that the fifth row of options has the value CONNECTED highlighted. That setting tells the calculator to fill in extra pixels in cases such as the Y=4X+1 shown above, where a pixel in one column is not touching (on the corner or on the side) the calculated pixel in the next column.
Figure 25	The alternate to CONNECTED is DOTS. We can move to select DOTS by pressing the key four times to move the cursor to that row of options, and then pressing the key to highlight the DOTS option. Figure 25 shows such a condition, with the blinking cursor covering the DOTS option.
Figure 26	To actually make the change we need to press the key. Figure 26 shows the option changed to DOTS.
Figure 27	We can return to the GRAPH screen by pressing the key. The new graph is shown in Figure 27. Note that the connecting pixels have not been turned on. A close examination will show that the pixels associated with the points we selected earlier, (0.3,2.2), (0.4,2.6), and (0.5,3.0), are exactly the pixels that have been turned black. To facilitate your examination, Figure 27 is reproduced below, but 3 times as wide and 3 times as tall.