Projecting Rational Function Graphs onto a Cylinder
Here we have a lovely family of rational functions.
Move the slider; notice the equation; explain what you can.
I hope your explanation includes the words "vertical asymptote".
To access the same graph, seen from a 3D point of view, please open this URL in another window:
https://www.geogebra.org/calculator/e7kgku6n
As you play with that applet, consider these ideas.
[ShowThings = 0]
I've duplicated the x-axis at a height of 1. Think of that as the rail of a tram (A) that travels over the landscape.
I've marked point B on the green curve and the line of sight from the tram to B.
The question is, could the infinite landscape beneath the tram be an illusion? If we were to surround the tram rail with a half pipe projection screen, what points on that projection screen would give a tram passenger the same visual experience that they would get from the graph of f(x) on the infinite landscape?
[ShowThings = 1]
Here's a half pipe. Notice that line AB always meets the half pipe in one point. How do you suppose it will look if we trace all of those points?
[ShowThings = 2 or 3]
When a=1, how close is each point on the projection screen to the point on the landscape that it mimics? (Not looking for a precise or even numerical answer here--I just want you to compare the curve on the plane and the curve on the half pipe.) Now try other values of a.
How does a vertical asymptote (from the plane curve) appear on the half pipe curve?
[ShowThings = 5]
We've closed up the half pipe projection screen to include the tram rail itself. Notice that line AB always meets the cylinder at the tram rail and one other point. How do you suppose it will look if we trace that other point?
[ShowThings = 6 or 7]
When a=1, how close is each point on the new projection screen to the point
on the landscape that it mimics? (Similar remark as before.) Now try other values of a.
How does a vertical asymptote from the plane curve appear on the cylinder curve?
[ShowThings = 8]
Simultaneous view of all curves and surfaces.
Now change f to any other rational function and play again. You may use the parameter a or not.
The End
P.S. Below, find an older version of the applet linked above.
- Advantage: I can embed it in this web page along with explanatory text.
- Disadvantage: it was made in GeoGebra 5, which does not support the locus feature in 3D views.