rational functions

[rober...@lcps.org]

I would like to show that the graph of f(x) = (x^2-4)/(x-2) has a
"hole" at (2,4). On the 83/84 I could accomplish this using a decimal
window but I can't figure out how to do it on the nspire. Any help
would be appreciated.

[John Losse]

The "hole" was a nice way to show students that the fraction and the "line" were the same except at x=2, but it didn't always work. (Consider y =(9x^2-1)/(3x+1)). That is, you had to jigger the window to match the function, and if the hole was at an irrational number, forget it.

I've tried to recreate this on the nspire by using a combination of window settings, line styles (discrete, dotted, etc.) and parametric mode. No luck, but maybe somebody else has something.

On the other hand, even though a hole does not appear in the graph of (x^2-4)/(x+2), when you go into Graph Trace on the nspire and manually enter x=2, you get (2, undef), and the little crosshairs tracing the curve disappear, which is a nice feature.

By the way in experimenting I ran into something strange:

Enter Robert's function parametrically as x1(t) = t, y1(t) = (t^2-4)/(t-2) and go into Graph Trace. Set the trace step to 0.1 and start tracing at, say, t=1. As you trace back and forth across the trouble spot t=2, the trace step changes from the 0.1 you specified. I have found this annoying in other contexts as well (such as polar).

John Losse

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[Marc Garneau]

Awhile ago I threw together the attached document to play around with the idea of having an indicator for hollow point situations and vertical asymptotes.  The user traces the function by moving the point on the axis in the lower screen.  I can't remember how much rigor I put into this, so there could likely be some situations that lead to undesirable results.  Here are a couple of screenshots as a preview:

Marc Garneau
 
  

[John Losse]

Marc's document "HollowPoint" does a great job of illustrating both the hole and the vertical asymptote in the style of the TI 83/84 (where the asymptote was an artifact of "connected".)

In experimenting with it, it did seem that, if you change the function to one where the hole or asymptotes are at, say, 1/2 and 2/3, some adjustments would have to be made to the slider setup to get the hollow point and asymptote to appear. I couldn't see how to make these adjustments, and would be interested in how Marc designed the document to achieve what he did, which is pretty slick.

So far, it seems that, as with the 83/84, it is possible to show the hole in the graph and the vertical asymptote on the nspire only if the "atmospherics" are just right.

John Losse

[Joe]

I've had good success by pointing out that because the hole is at
exactly one x value it has zero width and therefore you don't see it.

[Sean Bird]

Amen to that.

And I also agree with an earlier comment that the trace and ctrl T, table, features have also taken care of exploration at what is happening at a precise x-value.

[Marc Garneau]

With the document in its current state, in order to manage rational
discontinuities, 2 things would need to change:

For example, say I wanted to increment by 0.5...
- in the hollow() program, change the x1 definition to x1:=round(2*x1,0)/2
- in the lower screen on page 1.1, change the x-scale to 0.5

If I were to do this document again, I'd probably approach it a bit
differently, and I'd introduce an x-scale slider to control how the
value of x increments, or I'd just control x using a built-in slider and
then the increment could be easily adjusted.

Marc Garneau

[Travis Bower]

...sounds like someone is getting closer to a projeeeeeect......LOL

go for it Marc, we will cheer you on!  Think of what you could do in color and maybe use McCalla's graphing/notes combo from his advanced authoring.  No pressure, but make it rock.