*[This is a post I wrote for the Sine of the Times blog.]*

I have been playing around with an idea in Sketchpad which I have been calling a "tactile function". It is a function defined by independent points. The user can move the independent points, and the function responds. In other words, you can define a function by grabbing the graph and shaping it, rather than by writing an equation.

**Download the file: TactileFunctions_parametric_2.gsp**

When I started exploring this, I realized that functions are usually considered to be symbolic ideas that have graphical representations. The definition lives in the equation, and if we want the graph to have certain qualities, we must (from experience or with trial and error) design an equation that results in those desired effects. What if instead we designed the graph directly and let the equation be what it may?

I was reminded of Driscoll's Doing-Undoing Habit of Mind. If we change our perspective to flip an idea around, we might encounter good math and interesting connections.

So can we define a function from its graph? That is what a math teacher does who scribbles a curve on the board. For example, if I want to show a curve that increases and changes concavity twice, or a curve that has several different extreme values, I can quickly draw a curve that has those characteristics, but it would be a different task to think of an equation describing such a curve.

The example drawn with chalk or marker has some limitations. It is strictly graphical, and there is no easy way to get a full description of it; the function is the product of a complex motion that the teacher cannot precisely explain. Such a function cannot really be used for anything besides its own graph.

Sketchpad can let us construct functions that are well-defined mathematical objects controlled by casual motions.

Recently my calculus students were studying parametric curves. The graphs in this topic are difficult for students to produce and comprehend, and I wondered how one could explore this topic through tactile functions.

In this sketch, two such functions represent x and y, which are both functions of t. The parametric curve off to the side is the path of a point whose location is defined by (x(t), y(t)). Select and drag any of the eight points on each of the curves in order to redefine that function. Try to make the parametric curve into a particular shape by forming the x and y functions as you like. For example, make a pretzel, a circle, a star, or your favorite letter. Hint: Look at one direction at a time, let's say x first. Try to look at what the x-values do, and make the x(t) curve show that pattern.

Some student work:

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