Pixels and Step Size on the TI-85: A Technical Exercise

By Gary A. Harris

Department of Mathematics

Texas Tech University

Lubbock, Texas

g.harris@ttu.edu

Recall that to graph a function on the TI-85 you choose GRAPH from the keyboard, the y(x)= from the bar menu and define your function,

say y1=x^2.

Next EXIT this menu and chose RANGE from the bar menu and enter your range for the values of x and y, and the scale for each axis.

Say xMin=-3.15, xMax=3.15, xScl=1, yMin=-3.15^2, yMax=3.15^2,	and
yScl=1.

Finally choose GRAPH from the bar menu and set back and let the machine do its thing. The point of this exercise is to explore just how the TI-85 goes about drawing that nice little picture on its screen.

The TI-85 selects sample values for x between xMin and xMax inclusively, approximates via some mysterious "numerical process" the corresponding values of y(x), just x^2 in this case, and "plots the point (x,y(x))" by lighting a little spot, called a pixel, above the sample x value on the x-axis and across from a spot on the y-axis that it thinks is pretty close to representing the corresponding value for y(x). Then it connects the consecutive spots by "straight lines" in the DrawLine FORMT or just leaves them as they are in the DrawDot FORMT.

The method for approximating y(x) most surely depends on how we define the function y(x), but the way the machine chooses the sample values for x is always the same. It chooses them uniformly distributed, one for each pixel from xMin to xMax; that is to say, consecutive sample values are all the same distant apart. If we knew the number of sample points, pixels, on the x-axis then we could determine just how far apart they must be.

Ex 1. Suppose there were 11 sample points, all the same distance apart from xMin=0 to xMax =10, inclusive. How far apart must they be?

Ex 2. Suppose there were 100 sample points from -25 to 25, how far apart must they be?

Ex 3. Suppose we know that all the sample points are .5 units apart on the range xMin=0 to xMax=20. How many sample points are there including both xMin and xMax?

Ex 4. Experiment with TRACE from the GRAPH bar menu using the above function and range to determine the number of sample points (pixels) on the x-axis of your machine. Including xMin and xMax there are equally spaced sample points on the x-axis.(It's cheating to look in your manual.)

Ex 5. If we want to sample at every integer value for x, with 0 exactly in the center, we must choose xMin= and xMax= (Check out the range after using ZINT from the ZOOM menu.)

Ex 6. Can you guess the effect on the value of xMin and xMax from using ZDECM from the ZOOM menu? Explain?

Ex 7. If xMin=0 what is the largest value possible we can choose for xMax so that both x=6 and x=9 are sample points? Use TRACE to verify that 6 & 9 are sample points. Explain why this is the biggest value for xMax that works.

Ex 8. If xMin=0 what is the largest value possible we can choose for xMax so that both x=5 and x=9 are sample points?

Ex 9. If xMin =0 what is the largest value possible we can choose for xMax so that both of two positive integers x=n and x=m are sample points? (Of course the answer is in terms of some integer related to the integers n and m.)

Ex 10. Counting xMin and xMax there are an odd number of pixels on the x-axis of the TI-85. Explain why this guarantees that the center of the range of x values is always a sample point. (Recall that the center of the interval from xMin to xMax is given by the formula (xMin + xMax)/2 and the length of the interval is xMax - xMin.)