Piecewise Demo

Piecewise Demonstration on TI-83

This page is devoted to presenting, in a step by step fashion, the keystrokes and the screen images for doing a piecewise funfunction definition on a TI-83. The example used is

Figure 1	First we will observe a fact or two. Press then to get Figure 1. Note the up-arrow in the cursor block indicating that we are in the yellow shift mode.
Figure 2	Now, press the key to open the TEST menu shown in Figure 2. We want to choose the "greater than" sign, which is option 3. We do this by pressing the key. This will paste teh "greater than" sign into our expression.
Figure 3	Here we see that the "greater than" sign is in our expression. We complete that expression with the sequence and then use the to evaluate the expression. The calculator responds with the value 1. This demonstrates that the value of "TRUE" is evaluated as 1.
Figure 4	We start the next expression with the same keys.
Figure 5	Press the key to return to the TEST menu. This time we will select option 6, less than or equal. We do this by pressing the key.
Figure 6	Again, the symbol has been pasted into the expression. We complete the expression with and press to evaluate the expression. The calculator responds with 0. Thus, the value FALSE is evaluated to 0.
Figure 7	The key opens the Y= screen shown in Figure 7. We will use the fact that TRUE is 1 and FALSE is 0 along with the the facts that 1 times a value is that same value, 0 times a value is 0, and 0 plus a value is that same value to help enter our piecewise function.
Figure 8	We start with the expression *(x0)(x² + 1). Whenever x is less than or equal to 0, the first factor, (x0), will have the value 1 so the entire expression will just have the value (x² + 1)**. On the other hand, if x is greater than 0, the first factor will have the value 0 so the whole expression will have the value 0 and it will not change the value of anything that we add to it.
Figure 9	In Figure 9 we continue the definition of the piecewise function by adding the second portion as *(0x)(x5)(x-1). If x less than or equal to 0 then the first factor is 0 so the entire expression will be 0. If x is greater than 5 then the second factor is 0 so the entire expression will be 0. It is only in the case that x is greater than 0 and* x is less than or equal to 5 that both first factors will have the value 1, in which case the third factor (x-1) determines the value of this expression.
Figure 10	We complete the piecewise function in Figure 10 with the last term, *(x5)(3x-11). This term will be 0 for all values of x less than or equal to 5. For values of x greater than 5 the factor (x5) evaluates to 1 making the value of this third term be defined by the value of the second factor, (3x-11)**.
Figure 11	pressing the produces the graph of the piecewise function shown in Figure 11.
Figure 12	To complete this examination, we want to look more closely at the function for values of x between -2 and 8. We open the WINDOW screen, shown in Figure 12 by presing the key.
Figure 13	Figure 13 shows the changes we have made.
Figure 14	Pressing produces the graph in Figure 14. This looks OK, until you really look at it. There is a small problem near the origin. Here is a magnified version of that area, but with the questionable pixels in red: We cannot explain those pixels by looking at the piecewise function definition.
Figure 15	In an attempt to look at this more closely, we return to the WINDOW screen, via the key, and we make the changes shown in Figure 15. Then, press the key to move to Figure 16.
Figure 16	This is more perplexing because there are now extra pixels to the right and below the origin, highlighted in red in the following: and the offending pixels from Figure 14 have disappeared.
Figure 17	The cause of all of this can be traced back to our original MODE settings. Figure 17 has the MODE screen as the result of pressing the key. The fifth option from the top is the choice between Connected and Dot. In connected mode, the current setting, the TI-83 as it graphs points of the function, tries to connect those points by filling in the vertical pixels between the points. A more lengthy discussion of this is in the pixels web page. We can just illustrate the featue here by changing this setting and then returning to the graph.
Figure 18	For Figure 18 we have moved the highlight over the Dot option and then pressed the key.

Figure 19	Press the key to open the ZOOM window, the key to select the standard view, i.e., WINDOW settings, and key to reurn to the graph. The offending dots have gone away, but so has the connected nature of the graph.