This is going to be a fuzzy-wuzzy post maybe. But, I wanted to write down how much I loved reading through the math work of my Calculus students, who completed those wonderful rollercoasters!
In the end, they took a variety of approaches:
* Some first chose the boundaries x = k, and then wrote down the boundary conditions f'(k) and f(k). They then took a generic form of the next adjacent function g, shifted it over to make g(x-k), and then differentiated g(x-k) and set g'(k - k) = f'(k), and g(k - k) = f(k) to solve for constraints on the remaining parameters in the g equation. In other words, they first chose boundaries, then did transformations, then got the derivatives to match via a standard algebraic approach (which we eventually, at the end of the project, went through as a class in preparation for their quiz).
* Some other students were clever. They first played with functions centered around x = 0, for example f(x) = ax^3, or f(x) = ae^x. They did this because it was easy to manipulate just "a" and the x value to get a numerical derivative and general shape that they wanted. For example, if they wanted a downwards parabola that connects with a derivative of 3, they might first get y = -x^2, and then figure that at x = -1, y' = -2(-1), so y' = 2. So, they figured that if they change "a" to be -1.5, then y' = 3. Bam, they got a general shape and a derivative value to match what they wanted at the boundary. And then all they had to do was to transform
y = -1.5x^2 over and up to the boundary, which is an easy task.
* Some other students chose their boundaries LAST. They first placed the pieces of functions down loosely, then took derivatives of connected equations and set the derivative equations equal f'(x) = g'(x). In their graphing calcs they solved for the x value where this occurs, and used that as the boundary x value. After that, they just shifted the g function up or down to meet the other function in height as well.
* Yet some other students used the principles of turning points to help them connect pieces. They also used horizontal symmetry around a vertex to predict steepness at a future part of a curve, etc.
* Many groups had trouble with ending their rollercoasters with the same height AND derivative value as in the very beginning. To help them make their lives easier, I recommended that they use the vertex form y = a(x - h)^2 + k on both ends, setting k to be the eventual height they wish to reach. They then had to put in an (x, y) value from the other boundary, and solve for a and h as a system. Even in doing this, there were some clever kids who did some clever substitution in order to make it easier to solve a rational system, while other kids turned it into a quadratic-linear system and solved graphically. Loved - it!
I felt really inspired by all of their individuality on this project. Even though it took a few more classes than I would have liked, I felt that the learning -- and moreover, the OWNERSHIP -- made it totally worth it in the end. Yeah!!!