Here are some examples illustrating various phenomena which arise when a calculator graphically represents a function. The examples are created specifically for the TI-85 but in most cases can be adapted to other calculators. The discussion on the graphics display screen page suggests how these adaptations might be achieved and why we have specified the particular RANGE windows in the examples.

• Have the calculator draw the graph of the function f(x) = x^(2/3) on the RANGE window [-10,10]×[-10,10]. After clearing the screen, then draw the graph of g(x) = (x^2)^(1/3) on the same window. Notice that the calculator graphs are entirely different (the first doesn't have any points to the left of the y-axis) even though the two functions are one and the same. Of course the second picture is the accurate one. This example illustrates that the TI-85's routine for computing powers of negative numbers has serious limitations. One needs to be careful to avoid this limitation.

• Sketch the graph of y = 1/(x-3) on the RANGE window [-10,10]×[-10,10]. This graph should have a vertical asymptote at x=3 but the calculator gives a poor representation. The reason for this has to do with the manner in which the calculator draws graphs. The basic procedure is that it plots the points whose x-coordinates are -10 + 20i/126 as i ranges from 0 to 126, and then connects these points by straight lines to draw the picture. There are two main ways to improve the calculator's performance on this graph: (1) On the GRAPH FORMAT menu change "DRAWLINE" to "DRAWDOT". Then the graphical representation will just consist of the plotted points with no connecting lines. (2) Choose a different window such as [-6.3,6.3]×[-3.1,3.1] or [-1,7]×[-10,10] so that the point with x-coordinate 3 is plotted. (Or in this case, not plotted since 3 is not in the domain of the function.)

• Notice the difference between the pictures for y = (x-3)/(x-3) with the RANGE windows [-10,10]×[-10,10] and [-6.3,6.3]×[-3.1,3.1]

• To get an idea of the effect of compression ratio on the graphical output of the calculator, graph the line y = x and the hemi-circle y = Sqrt(16 - x^2) on successive RANGE windows
  [-5,5]×[-10,10], [-10,10]×[-10,10], [-15,15]×[-10,10], [-20,20]×[-10,10], [-25,25]×[-10,10], [-100,100]×[-10,10] .

  The compression ratios for the windows are successively about 3.3994, 1.6997, 1.1331, .8498, .6799, .1700.

• On the "ZSTD" window have the calculator graph
  y = x - .004 sin(.01/x) .

  Notice that the graph passes through the origin (0,0). From this picture we might guess that the slope of the tangent line at the origin is about +1. Now zoom in on the graph at the origin 4 or 5 times. Observe that it now appears that the slope of the tangent line at the origin is about -1.
  

  The Moral: How can we ever know that we have zoomed in far enough on a graph to get an accurate picture of what's going on? The answer is that we have to use an algebra/calculus analysis of the function at hand in conjunction with the calculator to be confident that we are finding accurate information.

• The function f(x) = 3x⁴ - 400x³ - 60 x²

  has one local maximum and two local minimums. The local maximum occurs at x = 0 and local minimums are roughly at x = -.1 and x = 100. The local maximum and the first local minimum can be seen nicely in the RANGE window [-.2,.1]×[-1,1], and the second local minimum is nicely framed by the window [-10,150]×[-1.1× 10^8,10^8]. However notice that there is no hope of having one window which can simultaneously display all three extreme points.

• Graph y = cos(100 Pi x ) where Pi = 3.1415... denotes the usual constant (the area inside a circle with radius 1). Look at the graph using each of the RANGE windows:
  [-12.6,12.6]×[-2,2],   [-12.5,12.7]×[-2,2],   [-12.5,12.5]×[-2,2] .

  Notice the big changes in the picture when we make these seemingly small changes in the RANGE. How can it be explained?
  Find a window that gives a good representation of the actual graph of this function.

• In GRAPH FORMAT choose the settings "DRAWDOT" and "AXESOFF". Then have the calculator graph
  y = sin( 100x + {0,1,2,3,4,5,6,7,8,9,10,11,12} )

  on the RANGE window [-10,10]×[-1,1]. Note: The curly brackets { and } can be found on the "LIST" menu on the TI-85. They indicate a "list", so here there are 13 different functions listed and the calculator is graphing them one at a time.

• Simultaneously graph y = cos(100 Pi x) and y = cos(100 Pi (x - .1)) on the TI-85 using the RANGE window [-6.3,6.3]×[-1,1].
  

  This example suggests a related question: Can you find one function and an appropriate window so that every one of the 8001 pixels on the TI-85 screen gets darkened when the function is graphed?

This document was created in September, 1996 
and last revised on August 15, 1998.