I shared this applet with a learner recently. This is an account of our conversation.
She started by playing with the sliders, and then made the following pattern, saying "That's nice!":
She played about a bit more, and then made this image (the green and red curves are overlapping):
After around 5 minutes, she said: "Well, that's about it."
I drew her attention to the yellow box, and said: "What questions do you have?"
She said that she was not sure about the effect of a and b on the green curve, that it changed the height, but that it was hard to tell how. She also noticed there was also a shift, and then said that "It is always pi apart". I asked her what she meant by this, she pointed to the zeros of the green curve. I expressed my genuine excitement at this observation, which seemed to lend an energy.
She turned her attention to the red curve, and was much clearer about the effect of changing k and c (stretching and shifting).
She then turned her attention back to the green curve, changing a and b according to some scheme I could not determine. I asked her what she was attending to, she pointed to the maximum (of the green curve), that perhaps it was following a path similar to the red curve. (I wondered whether the path of the maxima was a tan curve (?), but decided not to share it)
She then set b = 0, and suggested that changing a from positive to negative makes the curve "flip over". She then changed b to 0.1, and altered a again, then b = 0.2, and so on. This brought the awareness that it was not a simple flip, and she remarked: "b = 0 seems to be special."
She then said that changing a "pushes the graph up at pi", and that changing b "pushes it up at pi/2". I asked her what she was attending to, she was attending to the points on the green curve with x coordinate x = pi, and now x = pi/2.
She then set b = 1, and increased a slowly from 0, and made the following conjecture: "The coordinate (pi/2, 1) stays the same, but the coordinate with x = 0 moves up with a." She then noticed that the coordinate with x = pi/2 changes according to b.
A lovely conjecture, and something that I had not noticed, which eventually became: The green curve contains coordinates (0,a) and (pi/2,b). [Why?]
However, she realised that this would not be enough to be able to know the shape of the whole curve, as the maximum changed in unpredictable ways, increasing/decreasing and shifting left/right with changes in a and b.
I then asked: "How does changing a and b affect the height of the curve?" She replied: "They both take it up." I then asked: "Would you be able to work out the height if I gave you values for a and b?", to which the answer was no.
Finally, I asked her if she thought there was a red curve that would always overlap any green one, to which she replied "Yes, because both the curves are always pi apart, and you can change the heights."
That was all we had time for.
The observations that the zeroes of the green curve are "pi apart", and that they can overlap, are the first steps in realising that the curves can always be made to overlap, for some choice of k and c.
What pedagogical choices could you make from here? In particular, how might you guide the learner from here to the core (content) aim of this exercise, namely to realise that we can derive the values of k and c that will overlap any given values of a and b?
A good starting point might be to return to the two examples that she generated, shown above.
It will certainly be worth returning to the fact that, whilst ascertaining the affect of changing a and b is difficult, it is much more easy to determine the effects of changing k and c, providing a rationale for the transformation from f(x) to g(x): to be able to sketch functions of the form y = a.cos(x) + b.sin(x).
It will also be interesting to return to her conjecture: why does the green curve contain the coordinates (0,a) and (pi/2,b)?