Function Dilations My math teacher presented this problem to us in class. He admitted that it was a poorly worded problem, and that it was confusing. However, I am still having a great amount of trouble in understanding the problem.
Image of what we were presented in class
He stated that the correct answer was D. The way he explained it was as follows: you take the coefficient of cos to use as the factor of dilation from the x-axis, and you take the coefficient of x to use as the factor of dilation from the y-axis. Then, you go up 3 on the y axis, even though the answer clearly states to move in the positive direction on the x-axis. All the other stuff is pretty straightforward.
Here's where I get confused. I believe the answer is C. The vertical shift of the graph is 3 up, and, therefore, A, B, and D appear to be wrong by default. In addition, I believe that it is the RECIPROCAL of the coefficient of x that would be used for the factor of dilation from the y-axis. This seems obvious to me, as the greater the coefficient of x, the more the graph shrinks horizontally, and therefore the dilation factor would be a fraction. I thought it was pretty straightforward, as you can actually multiply the old points by the dilation factors to see if you're right, and my solution appeared to make sense.
I have been to this resource, in my quest to understand, and it seems to reinforce my theory. Can anybody explain this to me, or maybe even in a way that my math teacher will understand, as I think he is just trying to defend the answer key he has (which he has said was written by people he believes to be incompetent).
 A: As you point out, answers A, B and D all refer to a translation on the $x$ axis, which is obviously wrong: when you get from $y=-3\cos(2x)$ to $y=3-3\cos(2x)$ as you add $3$ to $y$, so it's a translation on the $y$ axis, and the answer must be C.
IMO, it's not a good question, because the most unobvious part for most students is probably the compression caused by the coefficient $2$ in $\cos (2x)$, and not a stretching as one might think, as can be seen on the plot below. And this part does not require any thinking by the student if you already have the answer by simpler means.

The blue curve is $\cos(x)$, the other two curves show the effect of a coefficient $2$ (red) or $1/2$ (green).
This is not really a part of your exercise, but since you had the answer anyway, I'll show you below another unobvious feature of functions: when you shift the argument, the curve moves in the opposite direction. Actually, it's not unrelated to your question, since your professor tells you the solution involves a translation on the $x$ axis (which is wrong).
In the plot below, the curve $y=\cos(x)$ is still red, while $y=\cos(x+\pi/4)$ is blue and $y=\cos(x-\pi/4)$ is green. You see, adding $\pi/4$ to $x$ make the curve translate by $\pi/4$ to the left.

Of course, you have to be careful: because of the periodicity of cosine, if you add $2\pi-\pi/4$, which is a positive number, it's equivalent to subtract $\pi/4$ and the curve will move to the right. Indeed, $\cos(x+2\pi-\pi/4)=\cos(x-\pi/4)$, and you get the green curve.
The reason is simply that when you plot the curve $y=\cos(x+2\pi)$, the original curve $y=\cos (x)$ has moved enough to the left that the a period is superimposed with the next one, and it looks like if nothing had moved at all.
