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http://teachmaths-inthinking.co.uk/activities/transforming-functions-the-stretch.htm
Students find transforming functions one of the most demanding topics in mathematics at this level. In particular, the effect a has on the function, f(x) for y=af(x) and y=f(ax) and understanding just what a stretch is. This activity documents the most succesful approach I've had with this topic. It approaches the idea from the angle of transforming shapes, looking at the effect on the co