Chapter 3: Section 3: Demo

This page provides a demonstration of a corrected graph for the function given in Q6 on page 171 of the text, and a demonstration of using the SHADE function on the TI-83, as described at the bottom of page 181 of the text.

Question 6 on page 171 poses the situation where

x = time worked, in months
s(x) = total salary recieved
s(x) = 4000x + 3000 The sketch of the function is given on the next page of the text. However, that sketch is given as a continuous function. We know from the problem that the salary jumps by $4,000 each month, and that it starts at $3,000. At any time during the first month, before the month is completed, the salary is $3,000. At any time from the start of the second second month, before the month is completed, the the salary is $7,000. Let us produce a graph that follows that pattern.

Figure 1	To start, we press the key to examine, and change if need be, the mode settings. In particular, note that the fifth setting has the Dot option selected.
Figure 2	Next we will check the WINDOW settings. Press the key to open that screen and change the values as required. The settings shown in Figure 2 will handle our requirements. The Xmin value is set to ^–.2, which will mean that the Y-axis is visible. The Xmax value is set to 9.2 so that there are 94 steps from the minimum through the maximum value. This means that each step, left or right, will correspond to a change of 0.1 (because there are 95 pixels across the screen). The Y values are less critical. The values in Figure 2 were chosen so that we can see the functional values for the range of X values.
Figure 3	In order to define our function we press the key to move to the Y= screen. The function definition is Y1=4000(int(X))+3000 where int(X) is a built-in function that returns the largest integer less than or equal to X.
Figure 4	Pressing the key will move to the graph screen. Notice the effect of having set the calculator to DOT mode in Figure 1. The function is graphed for each X value, but the calculator does not try to connect a point to the next graphed point.
Figure 5	Figure 5 shows the effect of shifting into TRACE mode by pressing the key. Here we can see that if X has the value 4.5 then the functional value is 19000.
Figure 6	We can use the key to move the trace point to the right. Even at 4.9 months, the salary remains at 19000.
Figure 7	Moving one more dot to the right illustrates the jump in salary resulting from completing another month of work.

Figure 7 concludes the demonstration of the function from page 171.

The end of this section deals with graphs of inequalities. At the very end the text suggests that we can use the SHADE command of the TI-83 to produce a graph for an inequality. The following Figures step us through doing just that.

Figure 8 The first thing that we need to do is to prepare the calculator. The SHADE command affects the graph screen, but it is not part of a normal function definition. Instead, we will give the SHADE command from the normal screen. However, we want to have a blank graph screen to display the result of the SHADE command.
In Figure 8 we have returned to the Y= screen and have cleared any existing functions.

Figure 9 We move to the ZOOM screen, by pressing , so that we can use the ZDecimal option to set all of the WINDOW settings with one keystroke.

Figure 10 Selecting that option by pressing the will not only set those values, but also open the graph screen. Then we press the keys to open the DRAW screen shown in Figure 10. Note that Shade is the 7^th option in the menu.

Figure 11 Pressing will paste Shade( onto the main screen, as it appears in Figure 11. The Shade command requires two functions, a function to give the low value and a function to give a high value. Shade will use the WINDOW settings and it will determine the low and high value for each value of X across the graph screen. Then, Shade will cause all pixels between the low and high value to become opaque (appear black). We need to complete the command that we have started with a low and a high value.

Figure 12 In Figure 12 we have completed the command by selecting the constant function ^–100 as the low value, and X as the high value. The ^–100 was an arbitrary choice, convenient in that it is clearly off the screen.

Figure 13 Having created the command in Figure 12, we press to perform the command. The calculator shifts to the graph screen and performs the command, creating the shading that corresponds to Y<X. We had to manufacture that relationship by choosing X as the high value and ^–100 as the low value for the Shade command.

Figure 14 Let us create a new graph that corresponds to the inequality y < ^–0.7X+1 We return to the main screen by pressing . Then we can recall the previous command by using the keys. At that point we can use the cursor keys to move back to the X and then we can enter our new command, as shown in Figure 14.

Figure 15 The key will perform the command. Note that the TI-83 merely draws the new shading on top of the old shading. When we had pressed the CLEAR key in Figure 14 we merely moved out of the graph screen; we did not clear the previous shading.

Figure 16 The keystrokes will return us to the DRAW menu shown in Figure 16.

Figure 17 We select ClrDraw from the menu of FIgure 16 by pressing the key. The calculator responds by moving to the graphs screen shown in Figure 17, with the previous shading removed.

Figure 18 Again, the key returns us to the main screen, shown in Figure 18.

Figure 19 We can recall the previous command via

Figure 20 Finally, press to perform that command. This will shift the calculator to the graph screen where we can see the graph of y < ^–0.7X+1 by itself.

Figure 8	The first thing that we need to do is to prepare the calculator. The SHADE command affects the graph screen, but it is not part of a normal function definition. Instead, we will give the SHADE command from the normal screen. However, we want to have a blank graph screen to display the result of the SHADE command. In Figure 8 we have returned to the Y= screen and have cleared any existing functions.
Figure 9	We move to the ZOOM screen, by pressing , so that we can use the ZDecimal option to set all of the WINDOW settings with one keystroke.
Figure 10	Selecting that option by pressing the will not only set those values, but also open the graph screen. Then we press the keys to open the DRAW screen shown in Figure 10. Note that Shade is the 7^th option in the menu.
Figure 11	Pressing will paste Shade( onto the main screen, as it appears in Figure 11. The Shade command requires two functions, a function to give the low value and a function to give a high value. Shade will use the WINDOW settings and it will determine the low and high value for each value of X across the graph screen. Then, Shade will cause all pixels between the low and high value to become opaque (appear black). We need to complete the command that we have started with a low and a high value.
Figure 12	In Figure 12 we have completed the command by selecting the constant function ^–100 as the low value, and X as the high value. The ^–100 was an arbitrary choice, convenient in that it is clearly off the screen.
Figure 13	Having created the command in Figure 12, we press to perform the command. The calculator shifts to the graph screen and performs the command, creating the shading that corresponds to Y<X. We had to manufacture that relationship by choosing X as the high value and ^–100 as the low value for the Shade command.
Figure 14	Let us create a new graph that corresponds to the inequality y < ^–0.7X+1 We return to the main screen by pressing . Then we can recall the previous command by using the keys. At that point we can use the cursor keys to move back to the X and then we can enter our new command, as shown in Figure 14.
Figure 15	The key will perform the command. Note that the TI-83 merely draws the new shading on top of the old shading. When we had pressed the CLEAR key in Figure 14 we merely moved out of the graph screen; we did not clear the previous shading.
Figure 16	The keystrokes will return us to the DRAW menu shown in Figure 16.
Figure 17	We select ClrDraw from the menu of FIgure 16 by pressing the key. The calculator responds by moving to the graphs screen shown in Figure 17, with the previous shading removed.
Figure 18	Again, the key returns us to the main screen, shown in Figure 18.
Figure 19	We can recall the previous command via
Figure 20	Finally, press to perform that command. This will shift the calculator to the graph screen where we can see the graph of y < ^–0.7X+1 by itself.