Natural Forms with Piecewise Graphs
Here is an idea for an activity for students in college algebra up through calculus to help with understanding transformations of graphs and domains of functions. One of the fun aspects of functions defined by algebraic rules is that one can use them to draw curves on a computer by making plots of their graphs. Adjusting the formula for the function adjusts the shape of the graph and exploring this dynamically on a computer can help the student make connections with the meaning of the symbols in the formula.
The photo above is a detailed image of a dragonfly wing. The perimeter of the wing does not pass the vertical line test, and so is not the graph of a function of the horizontal position on the screen. However, we can divide the perimeter of the wing into pieces (the top portion of the perimeter, and the bottom portion), each of which is the graph of a function. Each of these we might further break up from left to right by the location of critical points (in unofficial terms: where the curve levels off for a moment or where it has a corner).
This breaks the top portion of the perimeter into four pieces and the bottom portion into three pieces, each of which is the graph of a function of the horizontal distance across the screen. The leftmost piece of the top portion looks like a quarter of a circle stretched horizontally (i.e., a quarter of an ellipse). The next piece to the right looks like the right arm of a flattened out downward facing parabola. After that, its fairly flat for while and then the final piece looks like the right arm of a downward facing parabola. Similarly, the leftmost piece of the bottom part looks like a part of the graph of the square root function flipped over the horizontal axis. The last two segments appear to be portions of flattened parabolas.
Consider the following four functions: the upper unit semi circle , the parabola , the identity function , and the square root function . With transformations of graphs and restricting the domain, one can use these four building blocks to create reasonable facsimiles of the 7 pieces of the perimeter. The experimentation can be done quickly (and freely) using GeoGebra.
- Save the picture at the top of the post.
- Open GeoGebra and use the Insert Picture tool to insert the picture into the plane.
- Use the Pointer Tool to click and drag the picture so that the origin of the coordinate system is at the edge of the wingtip (for simplicity).
- Typing Function[-sqrt(x),-1,5] into the input bar will produce the graph of the function over the interval superimposed on top of the figure.
- Double-clicking the plotted curve brings up a dialog box with the formula you entered which you can modify to transform the graph. Changing the -1 and 5 will change the restriction on the domain.
- Playing around for a little bit using the other three functions as well, you can produce something like this:
The point is that picture provides a concrete goal for the transformations and there is a payoff for using the transformation and restricting the domains of the formulas to define the various pieces. For students who find the abstract symbolism off-putting, this kind of activity can sustain interest and pave a foundation for understanding the abstraction.
For some other natural forms which could provide interesting challenges, check out the results of the Olympus BioScapes International Digital Imaging Competition.