Teacher2Teacher Q&A #19085

Plotting points on a graph

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Nov 25, 2007 at 10:15:00
Subject: Re: Plotting points on a graph

Well, this may explain why I'm not an EXPERT, but here goes. I think the function of a graph is to get a "picture" of a relationship. The x values that are permitted, the y-values associated with them (assuming the traditional x-y coordinate plane) and a general idea of how they are related. If I know I'm graphing a continuous curve, I think I would NEVER fail to draw the smooth curve which generalizes the idea. I may never be able to graph all the points; certainly simple polynomial functions have domains that reach infinitely in both directions, so that would be impossible. But I can try to extend my sketch (and in the end, graphs are never exact, we can NOT make points or lines in the true geometric sense) so that it covers all the things that might be of "interest" (a term defined by the level of the drawer, the question which generated the construction of the graph, etc.). I may want to extend the x-axis far enough to include all the zeros and feel that a couple of arrows pointing off to infinity to indicate the end behavior, or maybe a curve dropping down to show a horizontal asymptote ends with an arrowhead to indicate "and so fourth" in a picture image. Of course as soon as I press "Send," I may think of a good time to NOT draw the "suggestion" of a complete curve, but for now I think I would be on the other side of the fence. This allows me to be dramatic when I teach my students about point discontinuities. "See, the missing point does NOT show up on the graphing calculator. It looks like a complete line, but if we KNOW to look, sure enough, there is no value for x = (whatever). And they learn to embellish the smooth curve with an open circle to suggest a missing point. I assure you (and the expert) that none of them believes that is the exact size of a point, but I'm not sure without the smooth points filled in on each side of the open circle, if I could ever make so clear an example that it is JUST the one point missing. Well, if the expert IS right, at least you have some company in your error. Sometimes when I don't like the expert commentary on Shuttle lift-offs or other such events, I turn the channel, and usually there is another expert with a totally different view, and I just keep looking until I find an expert I can live with. -Pat Ballew, for the T2T service

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