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Sample functionsThe graphs below are the PLOT Manager results for various functions. Click on a thumbnail to see the original graph.
Comparing the graph of a Step function, Trunc(x), drawn with the Plot Manager, an Excel table, and MathematicaAs it happens a function with a lot of sharp twists is difficult for a software program to plot. The step function, y=Trunc(x) has lots of sharp turns. The results of plotting the function three different ways is shown below. Note that the Plot Manager does much better than a more straightforward way, such as with the use of an Excel Table. In addition, the Plot Manager is comparable to the result of Mathematica, a much more sophisticated and expensive program. One might even argue that Mathematica fails to do a complete job since it misses the the two points (-5, -5) and the (5, 5) points altogether!
Comparing the graph of Tan(x) drawn with the Plot Manager and an Excel tableThe function y = Tan(x) is another demonstration of how the Plot Manager software does a better job of graphing functions than more common methods. In this case, the function has two asymptotes, at x = -p/2 and at x = p/2 respectively. Note that the x values correspond to x = -90° and x = 90°. The results with the Plot Manager and with an Excel data table are shown below.
The graph on the left clearly identifies the asymptotes as shown by the vertical lines at x = -p/2 and at x = p/2. In the chart to the right, the function does not clearly show that it goes to +¥ or to -¥. It is possible to improve the appearance of the graph which has asymptotes by reducing the range of the y-axis, as shown below.
In the two examples above, the y-axis is restricted to values between -20 and +20. Note that the result of the Excel table now looks much better. However, the asymptotic nature of the graph is still more easily noticed in the PLOT Manager output.
Tips on improving the graph of a difficult-to-graph function (such as a step function)While people would have no problem visualizing a step function such as y=Trunc (x), most software programs have difficulty plotting it. This is because there are so many 'sharp edges' (jumps) in the function. . In cases such as this, where we know something about the function's behavior, it helps the software if we give it additional guidance. Consider plotting the Trunc(x) function from x = -10 to x = 10 with the Plot Manager. Starting with 5 data points and ending with 100 yields the plot on the left below. Notice there are many instances where the software draws line at an angle rather than a vertical line.
Since we know about the discontinuities at each integer value, telling the software to start with 21 points and continuing until it has at least 200 yields the curve on the right. Notice that most, though not all, of the transitions that occur at integer values are are now drawn with vertical lines. Of course, since we changed the number of starting points and the number of ending points, can we tell which was responsible for the improvement? To help answer that question, we look at the same function, Trunc(x), plotted from x = -5 to x = 5 (as shown below).
On the left above, the software started with 11 data points and finished with 100, while on the right, it started with 11 and finished with 200 data points. Notice that there is no perceptible difference between the two graphs. Consequently, we can conclude that it was information about the 11 distinct x-values that helped it draw a better graph. Remember that we knew, from our knowledge of the function, that it changes in an abrupt fashion at each of the 10 non-zero integer values between -5 and +5 . |
