Compare y=Cos(x) and transformation HELP

[aisha]

How does the graph y=-3 \cos (2\theta+ \frac {\pi} {4}) +3 Differ from the graph of y=\cos\theta

I think this graph is different because it has a vertical stretch by a factor of 3 and is translated upwards 3 units but I am not sure how to rearrange this to find the horizontal translation and the horizontal compression/stretch

Could someone please help me out! THANKS

 

[kishtik]

A computer program can be helpful - enjoy changing parameters and understanding the corresponding changes of the graph.

 

[borisleprof]

Let's suppose that theta is replace by t.
in cos(2t + pi/4) can be written cos2(t+pi/8) that you can compare with cosa(t-h) where the parameter h gives you how many units the curve is translate to the right (when h >0) or to the left (when h<0)
Now you can see that in your problem h = -pi/8, so the translation is to the left pi/8 units. The parameter a = 2 tells you the number of times it osccillates in the intervall length of 2 pi; in the problem you have 2 complete oscillations in an intervall length of 2 pi.