For a polynomial expression y=f(x) find by a simple geometric mechanism the value of y for a given x, and have the mechanism trace out the curve represented by y=f(x).[br]Function: [math]y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6x^6[/math] [br][br]Von Sanden appears to have been the originator of a graphical method to perform this task which is here illustrated with the aid of GeoGebra. His approach is a geometrical adaption of Horner's earlier scheme, one that is principally based on simple addition to determine the value y=f(x) for a given x.[br][br]In Von Sanden's graphical model x and y are presented as a point on a correctly scaled graph. Variable x has the range 0<=x<=1. However by making the substitution t=x/L a new function y=g(t) may be formed, effectively extending the range to x <=L.[br][br]The graphical method is presented, with examples, in PRACTICAL MATHEMATICAL ANALYSIS by H. VON SANDEN, translated from the German into English by H. LEVY, 1923. See pages 54 - 60. The book may be viewed or downloaded from www.archive.org.[br][br]In this GeoGebra representation of Von Sanden's model the scale, for convenience, is distorted 1:10, but is alterable by the user.[br]The model as presented handles a polynomial up to power 6.[br][br]This model helps to bring home the fact that higher algebra was developed because at the time GeoGebra was a fantasy.

Using the sliders, set up the coefficients of the polynomial function y=f(x). Both + and - values are permitted, however Sliders have been set to a range -5 to +5 (alterable).[br]Varying slider "x" will cause the mechanism to alter and recalculate the value of y=f(x). The outcome is show geometrically as Point P(x,f(x)). Moving the slider over its full range traces out the curve of y=f(x).