At this model demonstrates a different number of roots of an equation depend on the equation parameter.
Consider equation |2(|x-1|-1)2-1|=a. To investigate a number of its roots see step-by-step construction of f(x) = |2(|x-1|-1)2-1| graph. Consequent transformations are x2, then to (x-1)2, then to (|x|-1)2, then to (|x-1|-1)2, then to 2(|x-1|-1)2, then to 2(|x-1|-1)2-1, and finally to |2(|x-1|-1)2-1|.
depending on a construct step-by-step a graph of f(x) = |2(|x-1|-1)2-1|. Transform x2 to (x-1)2 to (|x|-1)2 to (|x-1|-1)2 to 2(|x-1|-1)2 to 2(|x-1|-1)2-1 and finally to |2(|x-1|-1)2-1|. Now draw various horizontal lines. A cross point of a line and the graph is a root of the equation. Note that different lines may have different number of cross points (roots of the equation). At this manner there is no roots for negative a, 2 roots for a>1, 4 roots for a=0, 5 roots for a=1 and 8 roots are for 0<a<1.
The model is a movie. To navigate through the movie use Play, Pause, Stop, Previous step and Next step buttons.