Inverses of curves

A fellow Atlanta-area teacher and TI technology buff, Dennis Wilson, started talking to me several months ago about the power of using a locus to describe mathematical relationships. My own exposure to loci had been admittedly limited until that point. When I heard “locus”, my mind immediately went to geometry and the conic sections which I had “learned” by their locus definitions, but I really don’t think I had understood fully how useful a locus was.

The example I give below shows how to construct the inverse to any function using TI-Nspire software. The presentation uses computer software, but easily can be replicated on handheld Nspires. What I particularly like about this approach is that it gives a very strong visual representation of what had largely been a purely algebraic maneuver –switch x & y, and solve for y (if possible). I had always understood inverses as transformations in which the input and output variables reversed roles and knew that was equivalent to reflecting a given curve over y=x, but this approach significantly enhanced my understanding of locus and definitely improved my visual conception of the construction of an inverse.

What I’ve found most amazing is that this approach handles inverses of any curve you can graph in the function menu of the Nspire.

Finding graphs of inverses of non-functions has always been a challenging problem for computers. For example, to find the graph of the inverse of the ellipse using a function graph on an Nspire CAS, you could use the zeros command. Typically, this provides the entire graph, but this inverse approach results in only half of the ellipse and the corresponding half of its inverse. Because the zeros command has limitations for higher order polynomials, I hoped the approach would work for parametrically defined curves. The image below shows that the Geometry Trace still gives the outline of the curve (probably enough for most), but the locus doesn’t work. (I suspect this is because I’m creating the inverse from _dependent_ x- and y-coordinates.)

The same restrictions (probably for the same dependence-independence reasons) apply to polar curves, but at least you can trace the inverse.

I need to think more, but for now, thank you, Dennis, for opening a door in my mind that should have been opened long ago.


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