Chapter 32: Parametric Drawings: Situation 26 From the MACMTL–CPTM Situations Project

This example, appearing in CAS-Intensive Mathematics (Heid & Zbiek, 2004), was inspired by a student using a dynamic mathematics tool and mistakenly grabbing points representing both parameters (A and B in f (x ) = Ax + B) and dragging them simultaneously (the difference in value between A and B remains constant). This generated a family of functions that coincided in one point. Interestingly, no matter how far apart A and B were initially, if grabbed and moved together, the graphs of the functions in the family always coincided on the line x = -1.

In this case, dynamical geometry software was the vehicle that brought mathematical relationships to the fore. When one encounters such a phenomenon, one can enhance the experience by noticing the potential for mathematics in the patterns that are seen. Focus 1 uses transformations to explain the graphical phenomenon, whereas Focus 2 uses a symbolic proof. Focus 3 extends the phenomenon to quadratic functions (this discussion also appears in CAS-Intensive Mathematics). In addition, Focus 3 considers polynomials of higher degree (which generated another interesting relationship along with its proof). Focus 3 illustrates the decisions needed in designing an extension of a mathematical generalization.