Reasoning about Graphs of Functions
NOTE: the applets require Cinderella which can be downloaded free from Cinderella.de
ZigZagsWhat would the graph of f(x) = x – 2 – 1 look like?
What if the sequence [2, 1] was repeated over and over? What is the algebraic formula for the ZigZag shown? Characterise ZigZags that can be formed in this way. The Applet supports exploration of ZigZags. What about allowing multiplicative coefficients? What would be the same and what different for x – 2 – 1 and x – 2 – 1? 
MaxMin LinesHere you see three lines, each of which gives either the maximum value, or the minimum value on some interval. Can you do better?
What about n lines? The Applet supports up to six lines, and these can be adjusted geometrically (moving the white points) or algebraically (adjusting the coefficients of equations). 
Rational PolynomialsHere you see a quartic (green, numerator) and a quadratic (red, denominator). How much about the graph of their quotient can you deduce from their graphs?
The Applet supports polynomials up to degree 7. The notes provide information about the applet, and suggests ways of working with it. See also Mason (2015). Mason, J. (2015). Developing & Using an Applet to Enrich Students’ Concept Image of Rational Polynomials. Teaching Mathematics and its Applications. doi: 10.1093/teamat/hrv004 

CobWebsThe animation introduces a construction. What is happening? Starting from X, where does the white point end up and what is its locus?
The applet supports work on this question, and displays the locus. There are notes to help manage the applet. The applet also enables investigation of both horizontal and vertical alignments. 

Iterated CobwebsThe animation shows a phenomenon that needs explaining.
The applet permits other polynomial and control over the number of iterations, Some of the parameters are chosen by pressing and holding a button while typing in a number. 